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...

 

[m2003]

Hello,

Thank you for sharing your attempt at solving this problem. It looks like you have made some good progress so far. To help you understand the steps between your current solution and the final solution, let's break it down:

First, let's look at the equation you have derived:

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

This is a good start, but it's not quite the same as the final solution. To get there, we need to make a couple of substitutions. First, we can replace the "d" in the denominator with the distance between the two masses at that instant, which is represented as "d" in the final solution. This will give us:

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

Next, we can use the definition of the reduced mass, μ, which is given by μ = mM/(m+M). This allows us to rewrite the numerator in the equation as (m+M) = μ(m+M), giving us:

v^2= 2GM[μ(m+M)/d^2]

Finally, we can simplify this further by using the definition of the gravitational constant, G, which is given by G = 6.67 x 10^-11 Nm^2/kg^2. This allows us to rewrite the entire equation as:

v^2= 2Gμ(m+M)/d

Now, if we take the square root of both sides, we get the final solution:

v= sqrt(2Gμ(m+M)/d)

I hope this helps clarify the steps between your solution and the final solution. Keep up the good work!