Particle collides with hard wall When a particle elastically collides with a hard wall at some velocity, can the point at which the particle instantenously jumps be assumed to have zero velocity for potential to be continuous?
 A: Assuming a head-on collision (or only considering the normal component of the velocity), we could apply the mean value theorem to argue as follows: If velocity is given as $v(t) = dx(t)/dt$ where $x(t)$ is the distance from the wall at time $t$, and we assume that $x(t)$ is continuous and differentiable (i.e. changes smoothly), then according to the mean value theorem there will be a time $t_0$ where $v(t_0)=0$ as long as a collision takes place (i.e. there are times such that $v>0$ at one point and $v < 0$ at another).
If instead the velocity changes instantaneously upon collision, $v(t)$ is discontinuous and there will be no derivative of $x(t)$ defined at the moment of collision. This would mean that there is no point where the velocity is zero, but this would generally be considered a less accurate model.
A: In general it is better to think of the velocity of such a particle as discontinuous.
One of the most important conservation laws is the conservation of energy.  If, at any moment, the velocity of a particle is zero, then its kinetic energy is zero.  If its kinetic energy was non-zero beforehand, then that energy must have gone somewhere, and we typically need to identify where it went before the equations make any sort of sense.
Unfortunately, things talked of as "particles" rarely have a good method of storing such potential energy.  Distributed objects can store it in elastic bonds within their structure, but rarely do we give those properties to "particles."
In my experience, the idea of "collides with a hard wall" gives way before we start talking about a point where the velocity is zero.  In the real world, collisions are a complicated process with many intermediate steps as energy and momentum are smoothly transferred.  Objects are only talked about as "particles" when we can elide such details.
And, of course, real particles on the subatomic level don't "collide" with hard surfaces.  They interact electrostatically over the course of nanometers, and we find their path can indeed include a moment with zero velocity, where all of the energy is bundled up in electrostatic potentials.
