Central force motion and particles

[cardamom]

Homework Statement

The problem involves two particles of masses m and M; initially, m is at $$r=∞$$ and has a velocity $$v=v_o$$. The path of m is deflected, ie pulled towards M due to its gravitational pull.

Question: Find the mass M (in terms of the quantities given) at a distance d where the particles are now acting on each other.

Homework Equations

Initial energy of m

$$<br /> E_i = \frac{1}{2}mv_o^2$$

$$E_f = \frac{1}{2}μv_o^2 + U(r) = \frac{1}{2}μv_o^2 + (\frac{-GMm}{d})$$

$$\frac{1}{2}μv_o^2 = \frac{l^2}{2μd^2}<br /> <br />$$

The Attempt at a Solution

I've tried using combinations of the above, but in the end, I am not confident that I am correct in my assumptions of E_f, otherwise this would be an easy algebraic game. I also considered that E_f should include the effective potential, but at distance d, the two particles haven't yet crossed, though they are at a distance such that the vector between them is orthogonal to the path of m at that point. Any guidance is appreciated!

 

[cardamom]

Am I missing something obvious?

 

[NascentOxygen]

Is this a question that you made up?

What is mu? What is l (lower-case L)?

 

[cardamom]

No, it's on an assignment... mu=Mm/(M+m) and l is the magnitude of the angular momentum. I can use l since the direction of L is constant as it's only in one plane - yes?