...because judging from the level of physics knowledge on display in this thread, most of you are unable to find their own arse in darkness in the worst toilet in Scotland!

The quantity to conserve during a fully elastic bounce is _momentum_ (a vector of length m*v), and energy loss during the bounce is deduced using E=m*v*v*1/2 on the two objects left in movement after bounce and comparing with the energy that the objects had before bounce. Any lost energy can be assumed to go into deformation of the target's liver. Assume a frictionless surface and the target initially at standstill. Use the kinetic energy of the incoming bullet to make assumption about incoming bullet velocity. Do not consider Special Relativity corrections. You have 5 minutes.

Yeah, the idea of a bouncing bullet on paper-thin, deformable vests is insanly dumb. A bulletproof vest protects you because it doesn't deform a lot, only so much that it can transform the bullet's energy fully into heat and non-reversible deformation (keeping the momentum of course) but not so much that the target will die due to rupture of inner organs. If the vest is paper-thin, I surmise that it will deform a lot, giving no protection whatsoever, even if it's untearable.

And no, you CAN'T push someone over by hitting him with a bullet, even if he's standing on one foot. A bullet makes holes, it doesn't punch.