The impossibility of absolute rigidity during collisions

[kotreny]

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.

 

[A.T.]

My conclusion is that, given differentiable velocity, the colliding bodies cannot be perfectly rigid.

Yes, given differentiable velocity. If the spheres have some mass, an instantaneous velocity change would mean infinite forces.

 

[kotreny]

I thought about it, and realized that compression not only has to exist, it causes the force during a collision, or just about any mass-to-mass interaction you can think of. In fact, compression is inherent in the very nature of mass and it's ability to influence other masses. Pushing something would be theoretically impossible if your hand were completely rigid. Even atoms must be compressed to create any force.

I never thought about mass this precisely before. Is it something people are usually taught?

 

[mikelepore]

A similar deformation of supposedly rigid objects occurs with tension. Tie one end of a rope to a building. Use your hand or a machine to pull the rope with force F. The building will also pull the rope with force F, a force pointed the other way (which is why the rope doesn't accelerate.) How does the building apply a force to the rope? The molecular bonds in the building are deformed slightly, and behave as stretched springs.