ADVANCED PLATE TECTONICS - PLATE MOVEMENT
AND VELOCITY CALCULATIONS
1 COMPLEX NUMERICAL PROBLEMS
1.1 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
1. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
2. Sum the vectors: 𝑣total =𝑣1+𝑣2
3. Calculate components:
– 𝑣𝑥=15−6cos110°=17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
4. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
5. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.2 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
1. Set up the equation: Distance = Rate × Time
2. Rearrange to solve for Time: Time = Distance ÷ Rate
3. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
4. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.3 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
1. Horizontal component: 𝑣ℎ= 𝑣cos𝜃 =4cos30°=3.46 cm/year
2. Vertical component: 𝑣𝑣=𝑣sin𝜃 = 4sin30°= 2 cm/year
3. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°=115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.4 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
1. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
2. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.5 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
1. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
2. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.6 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
1. Set up a coordinate system with the A-B boundary along the x-axis
2. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
3. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
4. Magnitude: |𝑣𝐶𝐴|=√42+1.52= 4.27 cm/year
5. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.7 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
1. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
2. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
3. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
4. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
5. Best-fit equation: 𝑑 =64.29𝑡−14.29
6. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.8 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
1. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
2. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
3. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
4. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.9 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
5. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
6. Sum the vectors: 𝑣total =𝑣1+𝑣2
7. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
8. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
9. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.10 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
10. Set up the equation: Distance = Rate × Time
11. Rearrange to solve for Time: Time = Distance ÷ Rate
12. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
13. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.11 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
14. Horizontal component: 𝑣ℎ= 𝑣cos𝜃 =4cos30°=3.46 cm/year
15. Vertical component: 𝑣𝑣=𝑣sin𝜃 = 4sin30°= 2 cm/year
16. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.12 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
17. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
18. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.13 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
19. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
20. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.14 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
21. Set up a coordinate system with the A-B boundary along the x-axis
22. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
23. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
24. Magnitude: |𝑣𝐶𝐴|=√42+1.52= 4.27 cm/year
25. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.15 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
26. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
27. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
28. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
29. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
30. Best-fit equation: 𝑑 =64.29𝑡−14.29
31. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.16 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
32. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
33. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
34. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
35. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.17 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
36. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
37. Sum the vectors: 𝑣total =𝑣1+𝑣2
38. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
39. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
40. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.18 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
41. Set up the equation: Distance = Rate × Time
42. Rearrange to solve for Time: Time = Distance ÷ Rate
43. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
44. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.19 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
45. Horizontal component: 𝑣ℎ= 𝑣cos𝜃 =4cos30°=3.46 cm/year
46. Vertical component: 𝑣𝑣=𝑣sin𝜃 = 4sin30°= 2 cm/year
47. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.20 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
48. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
49. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.21 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
50. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
51. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.22 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
52. Set up a coordinate system with the A-B boundary along the x-axis
53. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
54. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
55. Magnitude: |𝑣𝐶𝐴|=√42+1.52= 4.27 cm/year
56. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.23 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
57. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
58. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
59. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
60. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
61. Best-fit equation: 𝑑 =64.29𝑡−14.29
62. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.24 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
63. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
64. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
65. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
66. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.25 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
67. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
68. Sum the vectors: 𝑣total =𝑣1+𝑣2
69. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
70. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
71. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.26 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
72. Set up the equation: Distance = Rate × Time
73. Rearrange to solve for Time: Time = Distance ÷ Rate
74. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
75. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.27 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
76. Horizontal component: 𝑣ℎ= 𝑣cos𝜃 =4cos30°=3.46 cm/year
77. Vertical component: 𝑣𝑣=𝑣sin𝜃 = 4sin30°= 2 cm/year
78. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.28 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
79. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
80. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.29 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
81. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
82. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.30 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
83. Set up a coordinate system with the A-B boundary along the x-axis
84. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
85. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
86. Magnitude: |𝑣𝐶𝐴|=√42+1.52= 4.27 cm/year
87. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.31 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
88. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
89. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
90. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
91. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
92. Best-fit equation: 𝑑 =64.29𝑡−14.29
93. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.32 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
94. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
95. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
96. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
97. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.33 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
98. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
99. Sum the vectors: 𝑣total =𝑣1+𝑣2
100. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
101. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
102. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.34 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
103. Set up the equation: Distance = Rate × Time
104. Rearrange to solve for Time: Time = Distance ÷ Rate
105. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
106. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.35 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
107. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
108. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
109. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.36 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
110. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
111. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.37 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
112. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
113. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.38 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
114. Set up a coordinate system with the A-B boundary along the x-axis
115. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
116. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
117. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
118. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.39 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
119. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
120. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
121. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
122. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
123. Best-fit equation: 𝑑 =64.29𝑡−14.29
124. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.40 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
125. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
126. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
127. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
128. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.41 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
129. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
130. Sum the vectors: 𝑣total =𝑣1+𝑣2
131. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
132. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
133. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.42 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
134. Set up the equation: Distance = Rate × Time
135. Rearrange to solve for Time: Time = Distance ÷ Rate
136. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
137. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.43 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
138. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
139. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
140. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.44 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
141. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
142. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.45 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
143. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
144. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.46 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
145. Set up a coordinate system with the A-B boundary along the x-axis
146. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
147. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
148. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
149. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.47 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
150. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
151. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
152. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
153. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
154. Best-fit equation: 𝑑 =64.29𝑡−14.29
155. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.48 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
156. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
157. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
158. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
159. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.49 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
160. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
161. Sum the vectors: 𝑣total =𝑣1+𝑣2
162. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
163. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
164. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.50 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
165. Set up the equation: Distance = Rate × Time
166. Rearrange to solve for Time: Time = Distance ÷ Rate
167. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
168. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.51 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
169. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
170. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
171. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.52 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
172. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
173. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.53 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
174. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
175. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.54 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
176. Set up a coordinate system with the A-B boundary along the x-axis
177. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
178. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
179. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
180. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.55 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
181. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
182. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
183. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
184. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
185. Best-fit equation: 𝑑 =64.29𝑡−14.29
186. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.56 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
187. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
188. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
189. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
190. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.57 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
191. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
192. Sum the vectors: 𝑣total =𝑣1+𝑣2
193. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
194. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
195. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.58 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
196. Set up the equation: Distance = Rate × Time
197. Rearrange to solve for Time: Time = Distance ÷ Rate
198. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
199. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.59 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
200. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
201. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
202. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.60 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
203. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
204. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.61 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
205. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
206. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.62 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
207. Set up a coordinate system with the A-B boundary along the x-axis
208. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
209. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
210. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
211. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.63 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
212. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
213. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
214. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
215. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
216. Best-fit equation: 𝑑 =64.29𝑡−14.29
217. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.64 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
218. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
219. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
220. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
221. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.65 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
222. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
223. Sum the vectors: 𝑣total =𝑣1+𝑣2
224. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
225. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
226. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.66 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
227. Set up the equation: Distance = Rate × Time
228. Rearrange to solve for Time: Time = Distance ÷ Rate
229. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
230. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.67 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
231. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
232. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
233. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.68 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
234. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
235. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.69 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
236. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
237. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.70 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
238. Set up a coordinate system with the A-B boundary along the x-axis
239. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
240. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
241. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
242. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.71 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
243. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
244. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
245. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
246. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
247. Best-fit equation: 𝑑 =64.29𝑡−14.29
248. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.72 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
249. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
250. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
251. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
252. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.73 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
253. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
254. Sum the vectors: 𝑣total =𝑣1+𝑣2
255. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
256. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
257. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.74 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
258. Set up the equation: Distance = Rate × Time
259. Rearrange to solve for Time: Time = Distance ÷ Rate
260. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
261. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.75 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
262. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
263. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
264. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.76 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
265. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
266. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.77 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
267. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
268. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.78 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
269. Set up a coordinate system with the A-B boundary along the x-axis
270. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
271. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
272. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
273. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.79 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
274. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
275. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
276. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
277. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
278. Best-fit equation: 𝑑 =64.29𝑡−14.29
279. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.80 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
280. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
281. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
282. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
283. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.81 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
284. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
285. Sum the vectors: 𝑣total =𝑣1+𝑣2
286. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
287. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
288. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.82 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
289. Set up the equation: Distance = Rate × Time
290. Rearrange to solve for Time: Time = Distance ÷ Rate
291. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
292. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.83 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
293. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
294. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
295. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.84 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
296. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
297. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.85 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
298. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
299. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.86 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
300. Set up a coordinate system with the A-B boundary along the x-axis
301. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
302. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
303. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
304. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.87 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
305. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
306. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
307. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
308. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
309. Best-fit equation: 𝑑 =64.29𝑡−14.29
310. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.88 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
311. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
312. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
313. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
314. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.89 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
315. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
316. Sum the vectors: 𝑣total =𝑣1+𝑣2
317. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
318. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
319. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.90 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
320. Set up the equation: Distance = Rate × Time
321. Rearrange to solve for Time: Time = Distance ÷ Rate
322. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
323. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.91 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
324. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
325. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
326. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.92 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
327. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
328. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.93 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
329. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
330. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.94 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
331. Set up a coordinate system with the A-B boundary along the x-axis
332. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
333. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
334. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
335. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.95 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
336. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
337. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
338. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
339. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
340. Best-fit equation: 𝑑 =64.29𝑡−14.29
341. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.96 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
342. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
343. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
344. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
345. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.97 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
346. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
347. Sum the vectors: 𝑣total =𝑣1+𝑣2
348. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
349. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
350. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.98 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
351. Set up the equation: Distance = Rate × Time
352. Rearrange to solve for Time: Time = Distance ÷ Rate
353. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
354. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.99 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
355. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
356. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
357. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.100 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
358. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
359. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.101 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
360. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
361. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.102 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
362. Set up a coordinate system with the A-B boundary along the x-axis
363. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
364. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
365. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
366. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.103 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
367. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
368. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
369. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
370. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
371. Best-fit equation: 𝑑 =64.29𝑡−14.29
372. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.104 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
373. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
374. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
375. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
376. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.105 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
377. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
378. Sum the vectors: 𝑣total =𝑣1+𝑣2
379. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
380. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
381. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.106 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
382. Set up the equation: Distance = Rate × Time
383. Rearrange to solve for Time: Time = Distance ÷ Rate
384. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
385. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.107 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
386. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
387. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
388. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.108 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
389. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
390. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.109 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
391. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
392. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.110 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
393. Set up a coordinate system with the A-B boundary along the x-axis
394. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
395. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
396. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
397. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.111 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
398. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
399. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
400. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
401. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
402. Best-fit equation: 𝑑 =64.29𝑡−14.29
403. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.112 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
404. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
405. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
406. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
407. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.113 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
408. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
409. Sum the vectors: 𝑣total =𝑣1+𝑣2
410. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
411. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
412. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.114 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
413. Set up the equation: Distance = Rate × Time
414. Rearrange to solve for Time: Time = Distance ÷ Rate
415. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
416. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.115 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
417. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
418. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
419. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.116 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
420. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
421. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.117 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
422. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
423. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.118 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
424. Set up a coordinate system with the A-B boundary along the x-axis
425. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
426. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
427. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
428. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.119 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
429. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
430. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
431. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
432. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
433. Best-fit equation: 𝑑 =64.29𝑡−14.29
434. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.120 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
435. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
436. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
437. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
438. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.121 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
439. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
440. Sum the vectors: 𝑣total =𝑣1+𝑣2
441. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
442. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
443. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.122 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
444. Set up the equation: Distance = Rate × Time
445. Rearrange to solve for Time: Time = Distance ÷ Rate
446. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
447. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.123 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
448. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
449. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
450. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.124 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
451. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
452. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.125 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
453. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
454. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.126 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
455. Set up a coordinate system with the A-B boundary along the x-axis
456. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
457. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
458. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
459. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.127 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
460. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
461. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
462. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
463. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
464. Best-fit equation: 𝑑 =64.29𝑡−14.29
465. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.128 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
466. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
467. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
468. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
469. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.129 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
470. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
471. Sum the vectors: 𝑣total =𝑣1+𝑣2
472. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
473. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
474. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.130 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
475. Set up the equation: Distance = Rate × Time
476. Rearrange to solve for Time: Time = Distance ÷ Rate
477. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
478. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.131 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
479. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
480. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
481. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.132 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
482. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
483. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.133 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
484. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
485. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.134 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
486. Set up a coordinate system with the A-B boundary along the x-axis
487. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
488. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
489. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
490. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.135 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
491. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
492. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
493. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
494. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
495. Best-fit equation: 𝑑 =64.29𝑡−14.29
496. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.136 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
497. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
498. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
499. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
500. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.137 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
501. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
502. Sum the vectors: 𝑣total =𝑣1+𝑣2
503. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
504. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
505. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.138 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
506. Set up the equation: Distance = Rate × Time
507. Rearrange to solve for Time: Time = Distance ÷ Rate
508. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
509. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.139 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
510. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
511. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
512. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.140 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
513. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
514. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.141 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
515. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
516. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.142 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
517. Set up a coordinate system with the A-B boundary along the x-axis
518. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
519. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
520. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
521. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.143 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
522. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
523. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
524. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
525. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
526. Best-fit equation: 𝑑 =64.29𝑡−14.29
527. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.144 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
528. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
529. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
530. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
531. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.145 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
532. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
533. Sum the vectors: 𝑣total =𝑣1+𝑣2
534. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
535. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
536. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.146 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
537. Set up the equation: Distance = Rate × Time
538. Rearrange to solve for Time: Time = Distance ÷ Rate
539. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
540. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.147 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
541. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
542. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
543. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.148 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
544. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
545. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.149 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
546. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
547. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.150 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
548. Set up a coordinate system with the A-B boundary along the x-axis
549. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
550. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
551. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
552. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.151 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
553. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
554. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
555. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
556. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
557. Best-fit equation: 𝑑 =64.29𝑡−14.29
558. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.152 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
559. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
560. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
561. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
562. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.153 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
563. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
564. Sum the vectors: 𝑣total =𝑣1+𝑣2
565. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
566. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
567. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.154 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
568. Set up the equation: Distance = Rate × Time
569. Rearrange to solve for Time: Time = Distance ÷ Rate
570. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
571. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.155 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
572. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
573. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
574. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.156 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
575. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
576. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.157 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
577. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
578. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.158 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
579. Set up a coordinate system with the A-B boundary along the x-axis
580. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
581. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
582. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
583. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.159 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
584. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
585. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
586. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
587. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
588. Best-fit equation: 𝑑 =64.29𝑡−14.29
589. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.160 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
590. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
591. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
592. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
593. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.161 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
594. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
595. Sum the vectors: 𝑣total =𝑣1+𝑣2
596. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
597. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
598. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.162 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
599. Set up the equation: Distance = Rate × Time
600. Rearrange to solve for Time: Time = Distance ÷ Rate
601. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
602. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.163 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
603. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
604. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
605. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.164 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
606. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
607. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.165 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
608. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
609. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.166 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
610. Set up a coordinate system with the A-B boundary along the x-axis
611. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
612. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
613. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
614. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.167 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
615. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
616. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
617. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
618. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
619. Best-fit equation: 𝑑 =64.29𝑡−14.29
620. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.168 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
621. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
622. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
623. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
624. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
1.169 PROBLEM 1
Calculate the velocity of the Pacific Plate relative to the North American Plate given the
following data:
• Spreading rate at the East Pacific Rise: 15 cm/year
• Convergence rate at the Aleutian Trench: 6 cm/year
• Angle between the East Pacific Rise and the Aleutian Trench: 110°
Solution:
625. Convert spreading and convergence rates to vectors:
– East Pacific Rise: 𝑣1=15𝑖
– Aleutian Trench: 𝑣2=−6(cos110°𝑖+sin110°𝑗)
626. Sum the vectors: 𝑣total =𝑣1+𝑣2
627. Calculate components:
– 𝑣𝑥=15−6cos110°= 17.05 cm/year
– 𝑣𝑦=−6sin110°= −5.64 cm/year
628. Calculate magnitude: 𝑣 =√𝑣𝑥
2+𝑣𝑦
2=√17.052+(−5.64)2=17.95 cm/year
629. Calculate direction: 𝜃 =tan−1(−5.64/17.05)=−18.3°
The Pacific Plate moves at 17.95 cm/year relative to the North American Plate, at an angle of
18.3° south of east.
1.170 PROBLEM 2
A transform fault offsets a mid-ocean ridge by 350 km. If the half-spreading rate is 2.5
cm/year, how old is the oceanic crust at the transform fault?
Solution:
630. Set up the equation: Distance = Rate × Time
631. Rearrange to solve for Time: Time = Distance ÷ Rate
632. Convert units:
– Distance: 350 km = 3.5 × 10⁵ m
– Rate: 2.5 cm/year = 0.025 m/year
633. Calculate: Time = (3.5 × 10⁵ m) ÷ (0.025 m/year) = 1.4 × 10⁷ years
The oceanic crust at the transform fault is approximately 14 million years old.
1.171 PROBLEM 3
The Juan de Fuca plate is subducting beneath the North American plate at a rate of 4 cm/year.
If the subduction angle is 30° from horizontal, calculate:
• The horizontal component of the subduction velocity
• The vertical component of the subduction velocity
• The depth of the slab at a distance of 200 km from the trench
Solution:
634. Horizontal component: 𝑣ℎ=𝑣cos𝜃 = 4cos30°= 3.46 cm/year
635. Vertical component: 𝑣𝑣=𝑣sin𝜃 =4sin30°= 2 cm/year
636. Depth calculation:
– Angle of triangle: tan30°=depth/200 km
– Depth = 200tan30°= 115.47 km
The slab depth at 200 km from the trench is approximately 115.47 km.
1.172 PROBLEM 4
Two plates are diverging at a total rate of 5 cm/year. Magnetic anomalies show that the
distance from the ridge axis to the Jaramillo event (0.99 Ma) is 28 km on one side and 22 km
on the other. Calculate:
• The half-spreading rates for each plate
• The age of the oceanic crust 100 km from the ridge axis on the faster-spreading side
Solution:
637. Half-spreading rates:
– Plate 1: 28 km/0.99 Ma = 2.83 cm/year
– Plate 2: 22 km/0.99 Ma = 2.22 cm/year
638. Age calculation:
– Use the faster rate: 2.83 cm/year
– Time = Distance ÷ Rate
– 𝑇 =100 km/(2.83 cm/year)=3.53 million years
The oceanic crust 100 km from the ridge axis on the faster-spreading side is approximately 3.53
million years old.
1.173 PROBLEM 5
A hotspot track on a plate shows the following distances and ages:
• Volcano A: 0 km, 0 Ma
• Volcano B: 250 km, 5 Ma
• Volcano C: 600 km, 10 Ma
Calculate the plate velocity between each pair of volcanoes and discuss any changes in plate
motion.
Solution:
639. Velocity between A and B:
– Distance: 250 km
– Time: 5 Ma
– Velocity = 250 km / 5 Ma = 50 km/Ma = 5 cm/year
640. Velocity between B and C:
– Distance: 600 km - 250 km = 350 km
– Time: 10 Ma - 5 Ma = 5 Ma
– Velocity = 350 km / 5 Ma = 70 km/Ma = 7 cm/year
The plate velocity increased from 5 cm/year to 7 cm/year, suggesting an acceleration in plate
motion over time.
1.174 PROBLEM 6
A triple junction consists of three plates: A, B, and C. The boundary between A and B is a
transform fault with a slip rate of 4 cm/year. The boundary between B and C is a divergent
boundary with a full spreading rate of 3 cm/year. Calculate the velocity of plate C relative to
plate A.
Solution:
641. Set up a coordinate system with the A-B boundary along the x-axis
642. Velocities:
– 𝑣𝐵𝐴 =4𝑖 cm/year
– 𝑣𝐶𝐵 =1.5𝑗 cm/year (half-spreading rate)
643. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =𝑣𝐶𝐵 +𝑣𝐵𝐴 = 4𝑖+1.5𝑗 cm/year
644. Magnitude: |𝑣𝐶𝐴|=√42+1.52=4.27 cm/year
645. Direction: 𝜃 =tan−1(1.5/4)=20.6°
Plate C moves relative to plate A at 4.27 cm/year, 20.6° north of east.
1.175 PROBLEM 7
A seamount chain formed by a hotspot shows the following ages and distances from the active
volcano:
• Seamount 1: 100 km, 2 Ma
• Seamount 2: 300 km, 5 Ma
• Seamount 3: 600 km, 9 Ma
Using least squares regression, determine the best-fit plate velocity and the age of a seamount
450 km from the active volcano.
Solution:
646. Set up the linear regression equation: 𝑑 =𝑣𝑡+𝑏
647. Calculate sums:
– ∑𝑡 =16, ∑𝑑 =1000, ∑𝑡2=110, ∑𝑡𝑑 =5700, 𝑛 =3
648. Calculate slope (velocity):
– 𝑣 =𝑛∑𝑡𝑑−∑𝑡∑𝑑
𝑛∑𝑡2−(∑𝑡)2=64.29 km/Ma
649. Calculate y-intercept:
– 𝑏 =∑𝑑−𝑣∑𝑡
𝑛=−14.29 km
650. Best-fit equation: 𝑑 =64.29𝑡−14.29
651. For d = 450 km, solve for t:
– 450=64.29𝑡−14.29
– 𝑡 =7.22 Ma
The best-fit plate velocity is 64.29 km/Ma (6.43 cm/year), and a seamount 450 km from the
active volcano would be approximately 7.22 million years old.
1.176 PROBLEM 8
Three GPS stations on different plates form a triangle. Their relative velocities are:
• A relative to B: 5 cm/year at 030°
• B relative to C: 3 cm/year at 120°
• C relative to A: ?
Calculate the velocity of C relative to A.
Solution:
652. Convert velocities to vector components:
– 𝑣𝐴𝐵 =(5cos30°,5sin30°)=(4.33,2.5) cm/year
– 𝑣𝐵𝐶 =(3cos120°,3sin120°)=(−1.5,2.6) cm/year
653. Calculate 𝑣𝐶𝐴:
– 𝑣𝐶𝐴 =−(𝑣𝐴𝐵 +𝑣𝐵𝐶)
– 𝑣𝐶𝐴 =−(4.33+(−1.5),2.5+2.6)=(−2.83,−5.1) cm/year
654. Calculate magnitude:
– |𝑣𝐶𝐴|=√(−2.83)2+(−5.1)2=5.83 cm/year
655. Calculate direction:
– 𝜃 =tan−1(−5.1/−2.83)=209.0°
The velocity of C relative to A is 5.83 cm/year at 209.0° (or 29.0° west of south).
