Central force motion and particles

AI Thread Summary
The discussion centers on a physics problem involving two particles, one with mass m initially at infinity and moving towards another mass M under gravitational influence. The key equations include the initial and final energy expressions, which involve kinetic and potential energy terms. The participant expresses uncertainty about the effective potential energy at the distance d and whether their assumptions regarding the final energy E_f are correct. Clarifications are sought regarding the definitions of variables mu and l, with mu representing the reduced mass and l denoting angular momentum. The conversation highlights the complexities of central force motion and the need for precise understanding of energy conservation in the context of gravitational interactions.
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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 /> <br /> E_i = \frac{1}{2}mv_o^2<br />

E_f = \frac{1}{2}μv_o^2 + U(r) = \frac{1}{2}μv_o^2 + (\frac{-GMm}{d})<br />

\frac{1}{2}μv_o^2 = \frac{l^2}{2μd^2}<br /> <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!
 
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Am I missing something obvious?
 
Is this a question that you made up?

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