Those who demand that everyone "just ACCEPT that Oswald did it"...

[struggle4progress]

290. Rather simple physics calculations will show that matter can travel both forwards and backwards

after a collision

I'll do a simple version of the calculation, choosing units so all masses are measured relative to the mass of the bullet and all velocities relative to the velocity of the bullet. In particular the bullet mass is 1, and we can choose a coordinate frame so the bullet vector velocity is j and the object (of mass M) that the bullet strikes is initially at rest in the coordinate frame

My main simplifying assumption is that the bullet impact produces, from the bullet and the object is strikes, exactly two objects of masses P and Q (so P + Q = M + 1) by conservation of energy and that these objects travel only forward or backward along the original trajectory of the bullet so the velocities are pj and qj. This corresponds to the simplest possible breakup of the bullet combined with the object of mass M, and it will lead to a simple equality

It is clear we cannot have both p < 0 and q < 0: both final objects moving backwards would violate conservation of momentum. So let's assume p > 0. Can we ever have q < 0?

By conservation of momentum: j = (pP + qQ)j which implies pP + qQ = 1

By conservation of energy (all being kinetic here): 1 = ppP + qqQ

Thus p - q = p(pP + qQ) - q(pP + qQ) = ppP - qqQ

So (1) + {p - q} = (ppP + qqQ) + {ppP - qqQ} = 2ppP

or q = 1 + p - 2ppP

Recall that P > 0 and that we assume p > 0

The quadratic f(p) = 1 + p - 2ppP has positive root r = {1 + SQRT(8P + 1)}/(4P)

So q < 0 whenever p > r