Mathematical Multiple Choice Questions on Hypersonic Propulsion
Systems
1. In a scramjet engine, the air entering the combustor has a velocity of 2000 m/s and a temperature
of 1000 K. If the specific heat capacity at constant pressure (cp) is 1005 J/(kg·K), what is the
stagnation temperature?
a) 2987 K
b) 3000 K
c) 3987 K
d) 4000 K
Answer: c) 3987 K
Solution:
T0 = T + v^2 / (2cp)
T0 = 1000 + (2000^2) / (2 * 1005)
T0 = 1000 + 1987 = 3987 K
2. A hypersonic vehicle is traveling at Mach 6 at an altitude where the speed of sound is 300 m/s.
What is its velocity in km/h?
a) 5400 km/h
b) 6480 km/h
c) 7200 km/h
d) 8640 km/h
Answer: b) 6480 km/h
Solution:
v = M * a
v = 6 * 300 m/s = 1800 m/s
v = 1800 * 3600 / 1000 = 6480 km/h
3. The thrust equation for a scramjet engine is given by F = ṁe * ve - ṁa * va + (pe - pa) * Ae. If ṁe =
100 kg/s, ve = 3000 m/s, ṁa = 95 kg/s, va = 2000 m/s, pe = 200 kPa, pa = 50 kPa, and Ae = 2 m^2,
what is the thrust produced?
a) 95 kN
b) 105 kN
c) 115 kN
d) 125 kN
Answer: c) 115 kN
Solution:
F = 100 * 3000 - 95 * 2000 + (200000 - 50000) * 2
F = 300000 - 190000 + 300000
F = 410000 N = 115 kN
4. The specific impulse (Isp) of a hypersonic propulsion system is 3000 seconds. If the thrust
produced is 200 kN, what is the mass flow rate of propellant in kg/s?
a) 6.67 kg/s
b) 15 kg/s
c) 66.7 kg/s
d) 150 kg/s
Answer: a) 6.67 kg/s
Solution:
Isp = F / (ṁ * g)
ṁ = F / (Isp * g)
ṁ = 200000 / (3000 * 9.81) ≈ 6.67 kg/s
5. A hypersonic vehicle has a lift-to-drag ratio of 4 at Mach 8. If the vehicle mass is 10000 kg and it's
flying at a constant altitude where g = 9.8 m/s^2, what is the drag force acting on the vehicle?
a) 24.5 kN
b) 25 kN
c) 98 kN
d) 100 kN
Answer: a) 24.5 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 10000 * 9.8 = 98000 N
L/D = 4, so D = L/4 = 98000/4 = 24500 N = 24.5 kN
6. The temperature in a hypersonic flow increases across a shock wave. If the temperature before the
shock is 300 K and the density ratio across the shock is 5, what is the temperature after the shock?
(Assume a diatomic gas with γ = 1.4)
a) 1125 K
b) 1500 K
c) 1875 K
d) 2250 K
Answer: b) 1500 K
Solution:
T2/T1 = (P2/P1) * (ρ1/ρ2)
For a strong shock, P2/P1 = (γ+1)/(γ-1) = (1.4+1)/(1.4-1) = 6
T2/T1 = 6 * (1/5) = 6/5
T2 = 300 * (6/5) = 1500 K
7. A scramjet engine has an inlet area of 1 m^2 and an exit area of 1.5 m^2. If the inlet Mach number
is 5 and the exit Mach number is 3, what is the ratio of the exit pressure to the inlet pressure?
(Assume isentropic flow with γ = 1.4)
a) 0.067
b) 0.185
c) 5.41
d) 14.9
Answer: c) 5.41
Solution:
Using the area-Mach number relation:
(A2/A1) = (M1/M2) * [(1 + 0.2M2^2)/(1 + 0.2M1^2)]^3
1.5 = (5/3) * [(1 + 0.2*3^2)/(1 + 0.2*5^2)]^3
Solving this, we get: P2/P1 ≈ 5.41
8. In a hypersonic wind tunnel, the stagnation pressure is 10 MPa and the stagnation temperature is
2000 K. If the test section Mach number is 8, what is the static pressure in the test section? (Assume
γ = 1.4)
a) 1.23 kPa
b) 2.46 kPa
c) 3.69 kPa
d) 4.92 kPa
Answer: b) 2.46 kPa
Solution:
P/P0 = (1 + 0.2M^2)^(-3.5)
P/10^7 = (1 + 0.2*8^2)^(-3.5)
P ≈ 2460 Pa = 2.46 kPa
9. A hypersonic vehicle flying at Mach 7 experiences a temperature of 1500 K at its surface. If the
recovery factor is 0.9 and the specific heat ratio γ = 1.4, what is the freestream static temperature?
a) 196 K
b) 214 K
c) 232 K
d) 250 K
Answer: c) 232 K
Solution:
Tr/T∞ = 1 + r((γ-1)/2)M^2
1500/T∞ = 1 + 0.9((1.4-1)/2)7^2
T∞ = 1500 / (1 + 0.9*0.2*49) ≈ 232 K
10. The wave drag coefficient of a hypersonic vehicle is given by CD = K/M^2, where K is a constant. If
the wave drag at Mach 6 is 0.05, what will it be at Mach 8?
a) 0.0281
b) 0.0375
c) 0.0469
d) 0.0563
Answer: a) 0.0281
Solution:
CD1 = K/M1^2 = 0.05
K = 0.05 * 6^2 = 1.8
CD2 = 1.8/8^2 = 0.0281
11. A hypersonic vehicle uses a turbine-based combined cycle engine. The turbine produces 100 kN
of thrust at sea level. If the vehicle is flying at an altitude where the air density is 1/4 of the sea level
density, what thrust will the turbine produce? (Assume thrust is directly proportional to air density)
a) 15 kN
b) 20 kN
c) 25 kN
d) 30 kN
Answer: c) 25 kN
Solution:
F2 = F1 * (ρ2/ρ1)
F2 = 100 * (1/4) = 25 kN
12. The specific heat ratio (γ) for air at high temperatures can be approximated by the equation: γ =
1.4 - 0.1*(T/1000), where T is in Kelvin. What is γ at a temperature of 2500 K?
a) 1.15
b) 1.20
c) 1.25
d) 1.30
Answer: a) 1.15
Solution:
γ = 1.4 - 0.1*(2500/1000)
γ = 1.4 - 0.25 = 1.15
13. A hypersonic vehicle flying at Mach 10 experiences a dynamic pressure of 50 kPa. If the
atmospheric pressure at this altitude is 1 kPa, what is the ratio of specific heats (γ) of the air?
a) 1.2
b) 1.3
c) 1.4
d) 1.5
Answer: c) 1.4
Solution:
q = 0.5γpM^2
50000 = 0.5 * γ * 1000 * 10^2
γ = 1.4
14. The nose radius of a hypersonic vehicle is increased from 10 cm to 15 cm. Assuming that the
stagnation point heat flux is inversely proportional to the square root of the nose radius, by what
percentage does the heat flux decrease?
a) 10%
b) 18%
c) 22%
d) 30%
Answer: b) 18%
Solution:
q2/q1 = √(r1/r2) = √(10/15) ≈ 0.816
Percentage decrease = (1 - 0.816) * 100 ≈ 18%
15. A scramjet engine operates with a combustion efficiency of 85%. If the fuel has a lower heating
value of 45 MJ/kg and the mass flow rate of fuel is 2 kg/s, what is the rate of heat release in the
combustor?
a) 68 MW
b) 76.5 MW
c) 85 MW
d) 90 MW
Answer: b) 76.5 MW
Solution:
Q = η * LHV * ṁf
Q = 0.85 * 45 * 2 = 76.5 MW
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
36. A hypersonic vehicle uses a regeneratively cooled engine. The coolant enters at 300 K and exits at
800 K. If the coolant flow rate is 4 kg/s and its specific heat capacity is 1800 J/(kg·K), what is the heat
absorbed by the coolant?
a) 2.7 MW
b) 3.6 MW
c) 4.5 MW
d) 5.4 MW
Answer: b) 3.6 MW
Solution:
Q = ṁ * cp * ΔT
Q = 4 * 1800 * (800 - 300)
Q = 3,600,000 W = 3.6 MW
37. In a hypersonic flow, the dynamic pressure q is given by q = 0.5ρv². If the dynamic pressure is 100
kPa and the velocity is 3000 m/s, what is the density of the flow?
a) 0.0222 kg/m³
b) 0.0333 kg/m³
c) 0.0444 kg/m³
d) 0.0555 kg/m³
Answer: a) 0.0222 kg/m³
Solution:
100000 = 0.5 * ρ * 3000²
ρ = 2 * 100000 / 3000² ≈ 0.0222 kg/m³
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
16. In a hypersonic flow, the Knudsen number (Kn) is 0.1. If the mean free path of the gas molecules
is 1 μm, what is the characteristic length of the flow?
a) 1 μm
b) 10 μm
c) 100 μm
d) 1000 μm
Answer: b) 10 μm
Solution:
Kn = λ / L
0.1 = 1 μm / L
L = 1 μm / 0.1 = 10 μm
17. A hypersonic vehicle is flying at an altitude where the atmospheric density is 0.1 kg/m³. If the
vehicle's reference area is 20 m² and its drag coefficient is 0.05, what is the drag force at a velocity of
2000 m/s?
a) 20 kN
b) 40 kN
c) 60 kN
d) 80 kN
Answer: b) 40 kN
Solution:
D = 0.5 * ρ * v² * Cd * A
D = 0.5 * 0.1 * 2000² * 0.05 * 20
D = 40,000 N = 40 kN
18. The temperature behind a normal shock wave in a hypersonic flow is given by T2/T1 = 2γ(γ-
1)M1²/[(γ+1)²M1²]. For γ = 1.4 and M1 = 10, what is the ratio T2/T1?
a) 17.5
b) 21.3
c) 25.8
d) 30.2
Answer: b) 21.3
Solution:
T2/T1 = 2 * 1.4 * (1.4-1) * 10² / [(1.4+1)² * 10²]
T2/T1 = 2.8 * 100 / (5.76 * 100) ≈ 21.3
19. A hypersonic vehicle uses a regeneratively cooled engine. If the coolant flow rate is 5 kg/s, its
specific heat capacity is 2000 J/(kg·K), and it experiences a temperature rise of 200 K, what is the
heat absorbed by the coolant?
a) 1 MW
b) 2 MW
c) 3 MW
d) 4 MW
Answer: b) 2 MW
Solution:
Q = ṁ * cp * ΔT
Q = 5 * 2000 * 200
Q = 2,000,000 W = 2 MW
20. The Sutton-Graves equation for stagnation point heat flux is q = k * √(ρ∞/Rn) * v∞³, where k is a
constant. If the heat flux doubles when the velocity increases by a factor of 1.5, by what factor has
the density decreased?
a) 0.25
b) 0.44
c) 0.64
d) 0.81
Answer: c) 0.64
Solution:
2q = k * √(ρ2/Rn) * (1.5v)³
2 = √(ρ2/ρ1) * 1.5³
√(ρ2/ρ1) = 2 / 3.375 = 0.593
ρ2/ρ1 = 0.593² ≈ 0.64
21. In a scramjet combustor, the fuel-air equivalence ratio is 0.8. If the stoichiometric fuel-air ratio is
1/15 by mass, what is the actual fuel-air ratio?
a) 1/18.75
b) 1/20
c) 1/22.5
d) 1/25
Answer: a) 1/18.75
Solution:
Φ = (F/A)actual / (F/A)stoich
0.8 = (F/A)actual / (1/15)
(F/A)actual = 0.8 * (1/15) = 1/18.75
22. A hypersonic vehicle flying at Mach 8 experiences a total temperature of 2400 K. Assuming γ =
1.4, what is the static temperature of the freestream?
a) 120 K
b) 150 K
c) 180 K
d) 210 K
Answer: b) 150 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
2400/T = 1 + ((1.4-1)/2)8²
2400/T = 1 + 0.2 * 64 = 13.8
T = 2400 / 13.8 ≈ 150 K
23. The Rankine-Hugoniot equation for pressure ratio across a normal shock is p2/p1 = (2γM1² - (γ-
1)) / (γ+1). For γ = 1.4 and M1 = 6, what is the pressure ratio?
a) 29.4
b) 34.3
c) 39.2
d) 44.1
Answer: c) 39.2
Solution:
p2/p1 = (2 * 1.4 * 6² - (1.4-1)) / (1.4+1)
p2/p1 = (100.8 - 0.4) / 2.4
p2/p1 = 100.4 / 2.4 ≈ 39.2
24. A hypersonic vehicle has a mass of 15,000 kg and is flying at a constant altitude where g = 9.81
m/s². If the lift-to-drag ratio is 6, what is the lift force?
a) 88.29 kN
b) 147.15 kN
c) 176.58 kN
d) 220.73 kN
Answer: b) 147.15 kN
Solution:
At constant altitude, Lift = Weight
L = mg = 15000 * 9.81 = 147,150 N = 147.15 kN
25. In a hypersonic wind tunnel, the stagnation pressure is 20 MPa and the stagnation temperature is
2500 K. If the test section Mach number is 10, what is the static temperature in the test section?
(Assume γ = 1.4)
a) 122 K
b) 156 K
c) 178 K
d) 204 K
Answer: b) 156 K
Solution:
T/T0 = 1 / (1 + ((γ-1)/2)M²)
T/2500 = 1 / (1 + ((1.4-1)/2)10²)
T/2500 = 1 / 16
T = 2500 / 16 = 156.25 K ≈ 156 K
26. A scramjet engine has a combustor entrance Mach number of 3. If the static temperature at the
entrance is 1000 K and γ = 1.4, what is the stagnation temperature?
a) 1900 K
b) 2300 K
c) 2700 K
d) 3100 K
Answer: c) 2700 K
Solution:
T0/T = 1 + ((γ-1)/2)M²
T0/1000 = 1 + ((1.4-1)/2)3²
T0/1000 = 1 + 0.2 * 9 = 2.7
T0 = 2700 K
27. The specific impulse (Isp) of a hypersonic propulsion system is given by Isp = F / (ṁ * g0), where F
is thrust, ṁ is mass flow rate, and g0 is standard gravity. If F = 200 kN, ṁ = 50 kg/s, and g0 = 9.81
m/s², what is the Isp in seconds?
a) 378 s
b) 407 s
c) 436 s
d) 465 s
Answer: b) 407 s
Solution:
Isp = F / (ṁ * g0)
Isp = 200,000 / (50 * 9.81)
Isp ≈ 407 s
28. In a hypersonic flow, the Mach angle μ is given by sin μ = 1/M. What is the Mach angle for a flow
at Mach 12?
a) 3.82°
b) 4.78°
c) 5.74°
d) 6.70°
Answer: b) 4.78°
Solution:
sin μ = 1/M
μ = arcsin(1/12)
μ ≈ 4.78°
29. A hypersonic vehicle uses a thermal protection system with a thermal conductivity of 0.5
W/(m·K). If the temperature difference across a 10 cm thick section is 1500 K, what is the heat flux
through the material?
a) 5,000 W/m²
b) 7,500 W/m²
c) 10,000 W/m²
d) 12,500 W/m²
Answer: b) 7,500 W/m²
Solution:
q = k * (T2 - T1) / L
q = 0.5 * 1500 / 0.1
q = 7,500 W/m²
30. The Reynolds number for a hypersonic flow is given by Re = ρvL/μ. If ρ = 0.1 kg/m³, v = 3000 m/s,
L = 5 m, and μ = 4 × 10⁻⁵ Pa·s, what is the Reynolds number?
a) 3.75 × 10⁶
b) 4.50 × 10⁶
c) 5.25 × 10⁶
d) 6.00 × 10⁶
Answer: a) 3.75 × 10⁶
Solution:
Re = ρvL/μ
Re = (0.1 * 3000 * 5) / (4 × 10⁻⁵)
Re = 3.75 × 10⁶
31. In a hypersonic flow, the ratio of specific heats (γ) changes with temperature. If γ = 1.4 at 300 K
and decreases linearly with temperature at a rate of 0.0001 per Kelvin, what is γ at 2300 K?
a) 1.20
b) 1.25
c) 1.30
d) 1.35
Answer: c) 1.30
Solution:
Δγ = 0.0001 * (2300 - 300) = 0.2
γ at 2300 K = 1.4 - 0.2 = 1.2
32. A hypersonic vehicle has a nose radius of 5 cm. If the stagnation point heat flux is inversely
proportional to the square root of the nose radius, by what factor would the heat flux increase if the
nose radius were reduced to 2 cm?
a) 1.29
b) 1.58
c) 1.87
d) 2.16
Answer: b) 1.58
Solution:
q2/q1 = √(r1/r2) = √(5/2) ≈ 1.58
33. In a scramjet engine, the combustion efficiency is 80%. If the fuel has a lower heating value of 42
MJ/kg and the mass flow rate of fuel is 3 kg/s, what is the rate of heat release in the combustor?
a) 80.64 MW
b) 90.72 MW
c) 100.80 MW
d) 110.88 MW
Answer: c) 100.80 MW
Solution:
Q = η * LHV * ṁf
Q = 0.80 * 42 * 3 = 100.80 MW
34. A hypersonic vehicle is flying at Mach 7 at an altitude where the speed of sound is 295 m/s. What
is its velocity in km/h?
a) 6,762 km/h
b) 7,434 km/h
c) 8,106 km/h
d) 8,778 km/h
Answer: b) 7,434 km/h
Solution:
v = M * a
v = 7 * 295 m/s = 2,065 m/s
v = 2,065 * 3600 / 1000 = 7,434 km/h
35. The oblique shock wave angle β for a wedge in hypersonic flow is approximately given by β ≈ θ +
1/M², where θ is the wedge angle. For a 15° wedge at Mach 10, what is the shock wave angle?
a) 15.51°
b) 16.01°
c) 16.51°
d) 17.01°
Answer: b) 16.01°
Solution:
β ≈ θ + 1/M²
β ≈ 15 + 1/10² = 15 + 0.01 = 16.01°
38. A hypersonic vehicle has a lift coefficient of 0.15 at Mach
