Question

A 3000 kg lunar lander is in orbit 80 km above the surface of the moon. It needs to move out to a 400 km -high orbit in order to link up with the mother ship that will take the astronauts home.

How much work must the thrusters do?

Answer 1

As the lander moves to a higher altitude so gravity will do negative work,

Therefore the work that the thrusters need to do is the change in total energy of the lander.

Total Energy = Kinetic Energy Gravitational Potential Energy
TE = ½mv² - GMm / r ----------------Equation A

Now we can find out the squared speed of a circular orbit by equating centripetal acceleration with gravitational acceleration:

v² / r = GM / r²
v² = GM / r

Now we substitute into Equation A:

TE = ( 0.5*mGM / r ) - ( GMm / r )
TE = - 0.5*GMm / r
∆TE =0.5*GMm( 1/r₁ - 1/r₂ ) = work

r = moon radius altitude
r₁ = 1,737,000m 80,000m = 1817000 m
r₂ = 1,737,000m 400,000m = 2137000 m

Now the work = ∆TE = ½GMm( 1/r₁ - 1/r₂ )

= ½( 6.673 × 10^-11 N(m/kg)² )( 7.3477 × 10^22 kg )( 3000kg )( 1/r₁ - 1/r₂ )

= ( 7.35 × 10^15 Nm² )[ 1 / ( 1817000 m ) - 1 / ( 2137000 m ) ]

= ( 7.35 × 10^15 Nm² )[ 8.24 × 10^-8 m^-1 ]

= 6.06 × 10^8 J