Question

A 4600 kg lunar lander is in orbit 40 km above the surface of the moon. It needs to move out to a 280 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

Gravity will do negative work as the lander moves to a higher altitude, where the increase in potential energy is greater than the decrease in kinetic energy ( of a circular orbit ).

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

The squared speed of a circular orbit can be found by equating centripetal acceleration with gravitational acceleration:

v² / r = GM / r²

v² = GM / r

Substituting into Equation A:

TE = ( ½mGM / r ) - ( GMm / r )

TE = - ½GMm / r

∆TE = ½GMm( 1/r₁ - 1/r₂ ) = work

r = moon radius altitude

r₁ = 1,737,000m + 40,000m = 1,777,000 m

r₂ = 1,737,000m + 280,000m = 2,017,000 m

work = ∆TE = ½GMm( 1/r₁ - 1/r₂ )

= ½( 6.673 × 10^-11 N(m/kg)² )( 7.3477 × 10^22 kg )( 4600 kg )( 1/r₁ - 1/r₂ )

= ( 1.127 × 10^16 Nm² )[ 1 / ( 1,777,000 m ) - 1 / ( 2,017,000 m ) ]

= 7.55 × 10^8 J