Let's say we have two perfectly rigid spheres. One is at rest and the other is moving toward it with some differentiable velocity. When they collide, the first sphere will start moving with infinitesimal velocity and the second will reduce its speed by an infinitesimal amount. But since the spheres can't go through each other or distort themselves at all, the respective accelerations must be instantaneous, or else there would be a short discrepancy in their velocities and the distance they each must travel. My conclusion is that, given differentiable velocity, the colliding bodies cannot be perfectly rigid. I'm teaching myself mechanics, so I don't know when professors remind their students of this. I think it's an interesting point that, in theory, all things must be elastic to some degree. This occurred to me just today though, and I would like to know if I'm wrong. If I'm not, please tell me who first stated it.