Question 1
Calculate the orbital velocity for a satellite in a circular Low Earth Orbit (LEO) at an altitude of 400 km.
Earth's radius is 6,371 km and its mass is 5.97 × 10²⁴ kg. Gravitational constant G = 6.67430 × 10⁻¹¹ m³/kg/s².
v = √(GM / r)
Orbital Velocity:
km/s
Question 2
A satellite is in a circular orbit at 500 km altitude. Calculate the orbital period in minutes.
Earth's radius = 6,371 km, Earth's mass = 5.97 × 10²⁴ kg, G = 6.67430 × 10⁻¹¹ m³/kg/s².
T = 2π√(r³ / GM)
Orbital Period:
minutes
Question 3
Calculate the altitude of a geostationary orbit. Earth's radius = 6,371 km,
Earth's mass = 5.97 × 10²⁴ kg, G = 6.67430 × 10⁻¹¹ m³/kg/s².
r = ∛(GMT² / 4π²)
Altitude:
km
Question 4
A satellite is moving from a 400 km circular orbit to a 800 km circular orbit using a Hohmann transfer.
Calculate the Δv required for the first burn (at perigee of the transfer orbit).
Earth's radius = 6,371 km, Earth's mass = 5.97 × 10²⁴ kg, G = 6.67430 × 10⁻¹¹ m³/kg/s².
Δv₁ = √(GM(2/r₁ - 2/(r₁ + r₂))) - √(GM/r₁)
Delta-V (first burn):
km/s
Question 5
For the same Hohmann transfer (400 km to 800 km), calculate the Δv required for the second burn (at apogee).
Δv₂ = √(GM/r₂) - √(GM(2/r₂ - 2/(r₁ + r₂)))
Delta-V (second burn):
km/s
Question 6
Calculate the escape velocity from Earth's surface.
Earth's radius = 6,371 km, Earth's mass = 5.97 × 10²⁴ kg, G = 6.67430 × 10⁻¹¹ m³/kg/s².
v_escape = √(2GM / r)
Escape Velocity:
km/s
Question 7
A launch vehicle provides a total Δv of 9.5 km/s. If the rocket's exhaust velocity is 3.2 km/s,
what is the required mass ratio (initial mass / final mass) according to the rocket equation?
Δv = v_e * ln(m₀ / m_f)
Mass Ratio:
(dimensionless)
Question 8
A satellite in geostationary orbit (altitude 35,786 km) needs to be moved to a graveyard orbit 300 km higher.
Calculate the Δv required for this maneuver.
Earth's radius = 6,371 km, Earth's mass = 5.97 × 10²⁴ kg, G = 6.67430 × 10⁻¹¹ m³/kg/s².
Δv = √(GM/r₂) - √(GM/r₁)
Delta-V:
m/s
Question 9
Calculate the orbital velocity of the Moon around Earth.
Average Earth-Moon distance = 384,400 km, Earth's mass = 5.97 × 10²⁴ kg, G = 6.67430 × 10⁻¹¹ m³/kg/s².
v = √(GM / r)
Moon's Orbital Velocity:
km/s
Question 10
A spacecraft is in a circular orbit around Earth at 300 km altitude.
Calculate the additional velocity needed to place it on a parabolic escape trajectory.
Earth's radius = 6,371 km, Earth's mass = 5.97 × 10²⁴ kg, G = 6.67430 × 10⁻¹¹ m³/kg/s².
Δv = √(2GM / r) - √(GM / r) = (√2 - 1) * √(GM / r)
Additional Velocity:
km/s