Centrifugal force dynamics


Science Advisor
I know it's relatively easy to show that the only possible orbits given an inverse square centripetal force relation are conic sections, of which the only stable orbits are ellipses. However, I was wondering what the situation would be given a centrifugal force (really a misnomer, since it wouldn't follow anything like a circular trajectory) but basically something in a situation like this:

Fixed positively charged particle and proton flying nearby.

In this situation the force is always anti parallel to the radius vector. Obviously there is no bound orbit for this situation, and I'm thinking the type of path is going to be highly sensitive to initial conditions, but before I waste time working over the problem I figure I'd ask if any of you have insight into it (or have done anything similar).

Due to the inverse square relationship, I believe it will still be a conic section - in this case, the only sensible one - a hyperbola.

In fact, a hyperbola makes good physical sense here. Far away, the thing moves in a straight line, then when it collides it has a large curvature, then it quickly gets back down to "asymptotic", straight-line behavior.

That's my guess.

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