Physics problem help please?

Question

An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 59.0 km. To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 29.0 m/s.

If nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

Answer 1

I wonder why physics textbooks feel compelled to make students deal with unit conversions in problems that are clearly artificial.  Anyway...

Unless the book gives you other figures, the Moon's radius is R = 1,740 km, and its mass is M = 7.35×10²² kg.  You should keep R and M in the equation as long as possible, in case the book does have separate values.

Proceed as follows.  First, find the linear velocity v in a circular orbit at a height of h = 59 km, which is a total radius of revolution of R+h.  You do this by equating the force of gravity on the spacecraft with the centripetal force on it (since gravity is the only thing supplying that centripetal force).  This gives

GMm/(R+h)² = mv²/(R+h)

where m is the mass of the spacecraft, which will cancel out, and G is the gravitational constant, G = 6.67×10¯¹¹ m³/(kg · s²).  Multiply both sides by (R+h)/m to get

v² = GM/(R+h)

This means that the initial kinetic energy of the spacecraft is

K = mv²/2 = GMm/[2(R+h)]

(Don't worry, the m will go away in the end.)  Next, find the difference in potential energy for the spacecraft between a height of 59 km (where it starts), and 0 km (where it crashes).  The formula for the gravitational potential is ø = -GM/r, for a distance r from the center of the Moon.  Therefore, the difference in potential energy is the mass times the difference in potential, or

P = m[GM/R - GM/(R+h)] = GMm[1/R - 1/(R+h)]

The kinetic energy of the spacecraft when it crashes will then be

K* = K+P = mv*²/2

where v* is the final velocity.  Multiply both sides by 2/m to get

v*² = 2(K+P)/m

v* = sqrt[2(K+P)/m]

and since you can cancel out the m, and you know everything else, you now have v*.  You just have to do the unit conversion.

EDIT: You must be very careful with your units.  Don't forget that G is in the mks system, while R and h are in km, not m.