Central Potential Repulsive Scattering

[ispivack]

Homework Statement

Consider two particles interacting via a repulsive central potential U(r) = k/r with k > 0. Find the minimal distance between particles, when one of them (with mass m1) is coming from infinity with initial velocity v0, and approaching an initially resting particle (with mass m2) with impact parameter ρ.

Relevant Equations

U(r) = k/r

I have one problem with this question that I've been struggling with. Initially, the total energy should be given by E =m1* v0^2/2 (as U goes to zero, and m2 is at rest). However, if we write r = r1 - r2, we get E = mu*rdot^2/2 + U_eff(r), U_eff(r) also goes to 0, where mu is the reduced mass. However, rdot^2 is exactly v0^2 when r is infinity, as rdot = r1dot - r2dot = r1dot, since m2 is at rest. This seems to imply that m1 = mu, which makes no sense. What am I missing? If I assume that the first energy I stated is incorrect, I have no trouble with the rest of the problem.

 

[Abhishek11235]

Simply because you have switched frames and illegally equated them. One frame is the 3rd observer,observing collision. In this frame,both particles are moving once they are sufficiently close to each other. Another frame is the frame of particle in which it is at rest and observing another particle coming towards him. These 2 frames coincide if the particle which was at rest, is too much heavy in which m=##\mu##.

 

[ispivack]

When the particle m1 is infinitely far away, neither particle will have a force acting on it. So, both equations hold for the frame of the third observor, no?

 

[hutchphd]

Why do you think this is true? If I only have a single particle is its KE the same for observers in various inertial frames? Manifestly not.

 

[Abhishek11235]

When the particle m1 is infinitely far away, neither particle will have a force acting on it. So, both equations hold for the frame of the third observor, no?

Succintly,If you follow derivation of energy term using reduced mass(c.f. Goldstein), you see,you have to go into the frame of centre of mass. Now,conservation of energy is "Conservation" law not invariant law. It need not be same in every frame!