Collision Processes in One Dimension

When you hit a baseball with a bat, the abrupt change in the motion of the ball is obvious. The bat applies a large force to the ball for a short period of time causing the ball to change its motion in a small amount of space. The ball collides with the bat while the bat collides with the ball. You are not aware that the ball applies much of a force to the bat, unless (see extended bodies) the collision occurs too close to your hands. Indeed, the force applied to the bat by the ball is equal and opposite, during each instant of time, to the force applied to the ball by the bat. The impulse given to the ball is equal and opposite to the impulse given to the bat. Hence, the change of momentum for the ball is equal and opposite to the change of momentum for the bat. While (literally) this is happening the ball is moving though a small displacement. Hence, the force, that is applied to the ball, is doing work on the ball while the force, that is applied to the bat, is doing work on the bat. The work done on the ball is not necessarily equal to the work done on the bat, because the displacement of the ball during the collision is not necessarily equal to the displacement of the bat during the collision. Hence, the change in energy of the ball may not be equal and opposite to the change in energy of the bat. Like most collisions, this one happens too fast for us to keep track of the details. Indeed, real collision processes usually involve much more than just the essential details. As a first approximation we will consider a scenario that will allow us to ignore complications such as:

the rotation of the colliding objects.
the internal energy of the objects, except in the most trivial way.
interactions between the colliding objects and the "outside" world, the objects will be isolated.

With these points in mind let's construct a scenario involving railroad cars on a horizontal track. The weight of each car is matched by the force of the track acting on the car. The rotational energies of the wheels are taken to be negligible. In short, with apologies to Plato, we will illuminate the essential details of a real process by examining the physics of an idealized process.

For each of the three examples described below calculate:

the accelerations of both railroad cars at each of the labeled locations,
the velocities of both railroad cars at each of the labeled locations,
and the displacements of both railroad cars as they travel between each of the labeled locations.

Two railroad cars are equipped with bumpers that respond to compressional forces by producing a 6000 newton force as they collapse and rebound. If the collision is not too violent and the the coupling device does not engage, then the collision will be totally elastic. During a totally elastic collision kinetic energy will be transfered from one body to another so that the total kinetic energy just before the collision will be the same as the total kinetic energy just after the collision. If the coupling device latches when the cars come together the collision will be inelastic. During an inelastic collision kinetic energy will be transfered from one body to another but the total kinetic energy just before the collision will not be the same as the total kinetic energy just after the collision. During all (totally elastic, semielastic, and inelastic) collisions the total momentum just before the collision will be the same as the total momentum just after the collision.