Distance of Closest Approach

mm8070

1. The problem statement, all variables and given/known data
A particle, unknown mass, has velocity v0 and impact parameter b. It goes towards a planet, mass M, from very far away. Find from scratch (? i'm not sure why it says from scratch), the distance of closest approach.


2. Relevant equations
I believe this equation is relevant: V_eff(r)=L²/2mr + V(r)


3. The attempt at a solution
I haven't attempted this problem because I have no idea what distance of closest approach is. I looked throughout my book and haven't found anything.

HallsofIvy

This question really doesn't make sense to me. The "distance of closest approach" is just what it says- the distance a which the particle is closest to the planet as it flies by. Of course, it it hit the planet, that would be 0. But to calculate such a thing you would have to compute its trajectory which would involve knowing not only its initial distance and speed but also it initial direction of travel. when you said "velocity [itex]v_0[/itex], is that a velocity vector? That would help buit then your formula would be adding a number ([itex]L^2/2mr[/itex]) to a vector (V(r)). In any case, I don't see how the "impact parameter" would be relevant if the partical does not "impact" the planet.

mm8070

Maybe i should have written the equation as U_eff(r) = (angular momentum)²/2mr² + U(r). Where U(r) is the potential energy. I also should have mentioned that part b says to use the section in my book about hyperbolas to show that the distance of closest approach is k/(ε + 1) where k and ε are some ridiculous constants that I'm certain would waste your time if I gave them to you. I'm sorry about that

vela

Draw a line through the center of the planet, parallel to v₀. The particle is a distance b from this line initially. Use this information to calculate the angular momentum L of the particle.

Once you have that, you can use energy considerations to figure out what the minimum value of r the particle can achieve is.