  We consider a picture in which the transition from a big crunch to a big bang
corresponds to the collision of two empty orbifold planes approaching each
other at a constant non-relativistic speed in a locally flat background
space-time, a situation relevant to recently proposed cosmological models. We
show that $p$-brane states which wind around the extra dimension propagate
smoothly and unambiguously across the orbifold plane collision. In particular
we calculate the quantum mechanical production of winding M2-branes extending
from one orbifold to the other. We find that the resulting density is finite
and that the resulting gravitational back-reaction is small. These winding
states, which include the string theory graviton, can be propagated smoothly
across the transition using a perturbative expansion in the membrane tension,
an expansion which from the point of view of string theory is an expansion in
{\it inverse} powers of $\alpha'$. We argue that interactions should be
well-behaved because the string coupling tends to zero at the crunch. The
production of massive Kaluza-Klein states should also be exponentially
suppressed for small collision speeds. We contrast this good behavior with that
found in previous studies of strings in Lorentzian orbifolds.
