Determining work of a 3 particle system

[Dusty912]

Homework Statement

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?

Homework Equations

conservation of energy
U_i + k_i=U_f +k_f
U=-(GmM)/r
K=(1/2)mv²
-------------------------
f=ma
-(GmM)/r²=ma
-(GM)/r²=a
-(GM)/r=ar
(can apply to h height)

r=radius of astroid
h=height above center
m=mass of rocket
M=mass of astroid
G=gravitational constant
a=acceleration due to asteroids gravity

The Attempt at a Solution

U_i + k_i=U_f +k_f
U_i + k_i=U_f + 0
-(GmM)/r +(1/2)mv_i²=-(GmM)/h
the m masses cancel
-(GM)/r +(1/2)v_i²=-(GM)/h
-ar +(1/2)v_i²=-ah
r-(1/2)(1/a)v_i²=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

 

[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.