Question

A 2690-kg spacecraft is in a circular orbit 2320 km above the surface of Mars.

How much work must the spacecraft engines perform to move the spacecraft to a circular orbit that is 4730 km above the surface?

Answer 1

As the spacecraft is moved to a higher orbit, the gravitational potential energy increases and the kinetic energy decreases.
Gravitational potential energy = -GMm/ r
Kinetic energy = ½ mv²

According to conservation of energy, the sum of the gravitational potential energy and kinetic energy must remain constant.

Work = final sum – initial sum

To determine the kinetic energy, we must determine the value of v² at the 2 distances.

Since the orbit is a perfect circle, the gravitation force, which hold the spacecraft in orbit, is the centripetal force.

Gravitation force = (GMm)/r²
Centripetal force = (mv²)/r
Centripetal force = Gravitation force
(mv²)/r = (GMm)/r²
Divide both sides by m
(v²/r) = (GM)/r²
Multiply both sides by r
v² = (GM)/r

Kinetic energy = ½ mGM/r
Gravitational potential energy = (GMm)/r
Sum of KE and GPE = ½mGM/rr + (-GMm)/ r
Sum of KE and GPE = (-½ mGM)/r
The only thing that changes is the value of r.
Sum of KE and GPE = (1/r)(-½ mGM)

Work = final sum – initial sum
The only thing that changes is the value of r.
Initial value of r = r₁, Final value of r = r₂

Final sum = (1/r₂) * (-½ * m * G * M)
Initial sum = (1/r₁) * (-½ * m * G * M)

Work = [(1/r₂) * (-½ * m * G * M)] – [(1/r₁) * (-½ * m * G * M)]

Work = (1/r₂ – 1/r₁) * (-½ * m * G * M)

G = 6.67 * 10^-11
M = mass of Mars = 6.42 * 10²³ kg
m = mass of spaceship = 2690 kg
r is the distance, in meters, from the center of Mars to the spacecraft
r = radius of Mars + Altitude of spacecraft
Radius of Mars =3.37 * 10⁶ m
Initial altitude = 2320 km = 2.32 * 10⁶ m
Initial value of r = 3.37 * 10⁶ + 2.32 * 10⁶ = 5.69 * 10⁶ m
Initial sum = (-½ * 2690 * 6.67* 10^-11 * 6.42 * 10²³) ÷ 5.69 * 10⁶
Final altitude = 4730 km = 4.73 * 10⁶ m
Final value of r = 3.37 * 10⁶ + 4.73 * 10⁶ = 8.1 * 10⁶ m
Final sum = (-½ * 2690 * 6.67* 10^-11 * 6.42 * 10²³) ÷ 5.23 * 10^6

Work = (1/8.1 * 10⁶ – 1/5.69 * 10⁶) * (-½ * 2690 * 6.67 * 10^-11 * 6.42 * 10²³)
Work = 3.012 * 10⁹ Joules