Satllite Crashing into the Moon

[aldofbg]

< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 50.0km . To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 20.0m/s
If nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

What I've done is used the formula for circular orbit to find initial velocity. v=sqrt(GM/R). I got the velocity and subtracted 20 m/s to account for lost speed. I then used potential and kinetic energy to set up an equation such
KEfinal=KEinitial+Uinitial using the formula U=GM/R. I got the wrong answer. I thought about it and decided that I had to account for Gravitational Potential energy at the surface of the moon. Thus KEfinal+Usurface=KEinitial+Uinital -> KEfinal=KEinitial+Uinitial-Usurface. My answer is 5785 km/h but the correct answer is 6060 km/hr. (if anyone wants to know I've disregarded mass of the satellite b/c it cancels out in the equation). I would appreciate any help. I've never posted here so please excuse my formatting.

Constants are M=7.35X10^22 kg
R=1.74X10^6 m
G=6.673x10^-11

 

[gneill]

You've almost got it. Find $\xi = \frac{v^2}{2} - \frac{GM}{r}$ which is the specific mechanical energy of the new orbit. Note that it's the sum of the per-unit-mass KE and PE, where the zero reference for PE is at infinity. For any unperturbed orbit the specific mechanical energy is a constant over the whole orbit.

Then you can find the the speed v at any distance along the orbit using the fact that $\xi$ is a constant.

 

[haruspex]

< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 50.0km . To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 20.0m/s
If nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

What I've done is used the formula for circular orbit to find initial velocity. v=sqrt(GM/R). I got the velocity and subtracted 20 m/s to account for lost speed. I then used potential and kinetic energy to set up an equation such
KEfinal=KEinitial+Uinitial using the formula U=GM/R. I got the wrong answer. I thought about it and decided that I had to account for Gravitational Potential energy at the surface of the moon. Thus KEfinal+Usurface=KEinitial+Uinital -> KEfinal=KEinitial+Uinitial-Usurface. My answer is 5785 km/h but the correct answer is 6060 km/hr. (if anyone wants to know I've disregarded mass of the satellite b/c it cancels out in the equation). I would appreciate any help. I've never posted here so please excuse my formatting.

Constants are M=7.35X10^22 kg
R=1.74X10^6 m
G=6.673x10^-11

Your method is fine but I get the book answer. Please post your working.