Kinematics

HW7 – Kinematics

Understand all the recitation questions before trying HW.

2 randomly selected questions will be graded for correctness.

Rest of the questions will be graded for completion.

1. Velocity of the particle is given by: 𝑣 = 4.5𝑡2 − 27𝑡 + 22.5 ft/s.

a. Determine the displacement from 𝑡 = 0 to 𝑡 = 6𝑠. (Ans: -27 ft)

b. Determine the distance travelled from 𝑡 = 0 to 𝑡 = 6𝑠. (Ans: 69 ft)

2. A particle is moving along a straight line such that velocity 𝑣 = −4𝑥2 m/s. Particle starts from the

position 𝑥 = 2 at 𝑡 = 0. Determine the velocity as a function of time. (Ans: 𝑣 = − 16

(8𝑡+1)2 m/s)

3. A particle is moving along a straight line such that acceleration 𝑎 = −4𝑣2 m/s. Particle starts with

velocity 𝑣 = 2 m/s at 𝑡 = 0. Determine the velocity as a function of time.

4. Acceleration of a ball thrown vertically up from the surface of the earth is 𝑎 = −𝑔 ( 𝑅2

(𝑅+𝑦)2 ) where 𝑔 is the

acceleration due to gravity on sea level, 𝑅 is the radius of earth, and 𝑦 is height from the surface of earth.

What minimum initial speed from the surface of earth will cause the ball to never fall back on earth?

(This is called escape velocity). (Ans: 𝑣 = √2𝑔𝑅 )

(Hint: This is the initial speed such that the final speed is 0 at the height 𝑦 = ∞ ).

5. The elevator starts from rest at the ground. Its maximum acceleration is 5 ft/s2 and it maximum deceleration

is 2 ft/s2. What is the minimum time in which you can reach 40 ft above the ground.

Draw the velocity time graph for this situation. (Ans: 7.48s)

6. The elevator starts from rest at the ground. Its moves up with acceleration of 14 ft/s2. At t=1s a bolt from the

ceiling loosens and falls. At what time 𝑡 will the bolt hit the elevator floor? Elevator is 3m from floor to ceiling.

Use 𝑔 = 10 m/s2. (Ans: 𝑡 = 1.5 s)

Hint: At 𝑡 = 1s the bolt is moving up with the same velocity as the elevator. But now it is free, i.e. it has free fall

acceleration of -10. Also note the floor on which the bolt is about to fall, is moving (accelerating) up all this time

towards the bolt.

7. Here is the velocity time graph of a vehicle.

Draw the acceleration time graph.

Draw the position time graph. (Assume position is 2m at t = 0).

Please try doing this without using calculus.

Try using slopes and areas. Show your calculations.

12 m/s

t = 30s t = 50s

8. Here is the acceleration time graph of a vehicle.

a. Draw the velocity time graph from 𝑡 = 0 to 𝑡 = 50𝑠.

(Assume velocity is 0m/s at t = 0).

b. Draw the position time graph from 𝑡 = 0 to 𝑡 = 50𝑠.

(Assume position is 0m at t = 0).

c. Draw the speed time graph from 𝑡 = 0 to 𝑡 = 50𝑠.

d. Draw the distance time graph from 𝑡 = 0 to 𝑡 = 50𝑠.

9. A train of length 100m is moving with constant speed of 10m/s. A second train of 100m on a parallel track is just

behind it (at 𝑡 = 0s). It is moving with 5m/s but is accelerating at 6m/s2. How much time will it take for the

second train to completely overtake the first train.

(Ans: 𝑡 = 9.04s)

10. A ball is thrown from point A and it hits the ground at point B after 2.5s.

Determine the speed 𝑣𝐴 and the angle 𝜃𝐴 at which the ball was thrown at point A.

(Hint: the x component of motion has zero acceleration. The y component of motion has acceleration −𝑔. )

(Ans: 𝑣𝐴 = 23.2 m/s and 𝜃𝐴 = 30.5 0)

t=0 s t=???? s

First Second

First Second

10 m/s

5 m/s, and 6m/s2

15 m/s2

t = 10 s t = 50s

-10 m/s2