# Determining work of a 3 particle system

1. Oct 9, 2016

### Dusty912

1. The problem statement, all variables and given/known data
on a spherical astroid with a radius of 500km and an acceleration due to gravity of 3.0m/s2
How far from the surface will a particle go if it leaves the asteroid’s surface with a radial speed of 1000m/s?
2. Relevant equations
conservation of energy
Ui + ki=Uf +kf
U=-(GmM)/r
K=(1/2)mv2
-------------------------
f=ma
-(GmM)/r2=ma
-(GM)/r2=a
-(GM)/r=ar
(can apply to h height)

h=height above center
m=mass of rocket
M=mass of astroid
G=gravitational constant
a=acceleration due to astroids gravity
3. The attempt at a solution
Ui + ki=Uf +kf
Ui + ki=Uf + 0
-(GmM)/r +(1/2)mvi2=-(GmM)/h
the m masses cancel
-(GM)/r +(1/2)vi2=-(GM)/h
-ar +(1/2)vi2=-ah
r-(1/2)(1/a)vi2=h
plugging in the values gives me
500000m+(1/2)(1/(3m/s2)(1000m/s2)2=h

5166666.667meters from the center of the astroid. so 166666.667 meters above its surface. Which is the incorrect answer. the right answer is 250000 meters above the surface.

I know this should be a pretty simple problem, Not too sure where I went wrong. Thanks ahead of time

2. Oct 9, 2016

### Dusty912

Okay so I realized that I cannot use mgh fro the U final because the acceleration due to gravity is different here. So I found the mass of the astroid. If I use U=-(GmM)/r and solve for h will I get the correct answer?

and btw the title is wrong for this post.