Solving Planetary Motion Homework: Showing v=sqrt(2G (M+m)/d)

[Zeth]

Homework Statement

Two masses, m and M, are initially at rest at a great distance from each other. The gravitational force between them causes them to accelerate towards each other. Using conservation of energy and momentum, show that at any instant the speed of one of the particles relative to the other is:

v=sqrt(2G (M+m)/d)


where d is the distance between them at that instant.

The Attempt at a Solution

I have the solution sheet but am stuck as to what happens between:

0 = -GMm/d +1/2m v^2(1+m/M)

and

v^2 = 2GM^2/d(M+m)

where v is v_m, but that's not relevant again till the end of the question.

what I've gotten the top to reduce to is

v^2= 2GM[(m+M)/dm]

Am I missing some math trick here?

 

[chaoseverlasting]

The answer should be v=sqrt(2G (M+m)/d)*M, and you've made a calculation mistake somewhere...