A space craft's speed due to gravitaitonal force

[chrishobo]

I have this question on a practice test I am using to study. Thanks for any assistance.

I believe it is a Energy Principle question

Homework Statement

A spacecraft is coasting towards mars. The mass of Mars is 6.4e^23 kg and its radius is 3400km. When the spacecraft is 7000km from the center of mars, the spacecraft s speed is 3000 m/s. Later, when the spacecraft is 4000 km from the center of mars, what is its speed?

(i think the following is unimportant, but here it is anyways:)
Assume that the effects of Mars two tiny moons, the other planets, and the sun are negligible. Precision is required to land on mars, so make an accurate calculation, not a rough, approximate calculation. Start from a fundamental principle.

Homework Equations

Energy Initial = Energy Final + work (both the spacecraft and planet are in the system, so work is 0)

Kinetic Energy = .5*mass*velocity^2

Potential energy = -G*mass1*MASS2/radius

The Attempt at a Solution

K_si+K_Mi+U_i+mc²+Mc²=K_sf+K_Mf+U_f+mc²+Mc²

Mc² cancels
mc² cancels
K_M has a near zero change, so it cancels

Leaving

K_si+U_i=K_sf+U_f

Plugging in I get

.5*m*(3000)²+(-6.7e^-11)*(m*(6.4e²³)/(7000)=.5*m*(v)²+(-6.7e^-11)*(m*(6.4e²³)/(4000)

The m's (mass of craft) should cancel out on both sides,

and solving for v i get 95904 m/s



The key for the practice test (which doesn't give any work) states 4.3e³ m/s

 

[ideasrule]

You've got a unit problem because you used m/s for v but km for r. Be sure to use only meters (you can't only use kilometers because 6.7e-11 has units of m^3/(kgs^2).

 

[ideasrule]

Also, it's unnecessary to include the rest mass energy or the kinetic energy change of the planet when doing problems like this. Just write Ki+Ui=Kf+Uf and everyone will understand what you mean.

 

[chrishobo]

sweet. i feel silly for overlooking that

glad i did that now and not on the test.

Thanks!