204	Week 4: Systems of Particles, Momentum and Collisions

4.5: Collisions

A “collision” in physics occurs when two bodies that are more or less not interacting (because they are too far apart to interact) come “in range” of their mutual interaction force, strongly interact for a short time, and then separate so that they are once again too far apart to interact. We usually think of this in terms of “before” and “after” states of the system – a collision takes a pair of particles from having some known initial “free” state right before the interaction occurs to an unknown final “free” state right after the interaction occurs. A good mental model for the interaction force (as a function of time) during the collision is the impulse force sketched above that is zero at all times but the short time t that the two particles are in range and strongly interacting.

There are three general “types” of collision:

•Elastic

•Fully Inelastic

•Partially Inelastic

In this section, we will first indicate a single universal assumption we will make when solving scattering problems using kinematics (conservation laws) as opposed to dynamics (solving the actual equations of motion for the interaction through the collision). Next, we will briefly define each type of collision listed above. Finally, in the following sections we’ll spend some time studying each type in some detail and deriving solutions where it is not too di cult.

4.5.1: Momentum Conservation in the Impulse Approximation

All collisions that occur rapidly enough to be treated in the impulse approximation conserve momentum even if the particles are not exactly free before and after (because they are moving in a gravitational field, experiencing drag, etc). The validity of the impulse approximation will be our default assumption in the collisions we treat in this course, and hence we will assume that all collisions conserve total momentum through the collision. That is, the total vector momentum of the colliding particles right before the collision will equal the total vector momentum of the colliding particles right after the collision.

Because momentum is a three-dimensional vector, this yields one to three (relevant) independent equations that constrain the solution, depending on the number of dimensions in which the collision occurs.

4.5.2: Elastic Collisions

By definition, an elastic collision is one that also conserves total kinetic energy so that the total scalar kinetic energy of the colliding particles before the collision must equal the total kinetic energy after the collision. This is an additional independent equation that the solution must satisfy.

It is assumed that all other contributions to the total mechanical energy (for example, gravitational potential energy) are identical before and after if not just zero, again this is the impulse approximation that states that all of these forces are negligible compared to the collision force over the time t. However, two of your homework problems will treat exceptions by explicitly giving you a conservative, “slow” interaction force (gravity and an inclined plane slope, and a spring) that mediates the “collision”. You can use these as mental models for what really happens in elastic collisions on a much faster and more violent time frame.

Week 4: Systems of Particles, Momentum and Collisions

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4.5.3: Fully Inelastic Collisions

For inelastic collisions, we will assume that the two particles form a single “particle” as a final state with the same total momentum as the system had before the collision. In these collisions, kinetic energy is always lost. Since energy itself is technically conserved, we can ask ourselves: Where did it go? The answer is: Into heat102!

One important characteristic of fully inelastic collisions, and the property that distinguishes them from partially inelastic collisions, is that the energy lost to heat in a fully inelastic collision is the maximum energy that can be lost in a momentum-conserving collision, as will be proven and discussed below.

Inelastic collisions are much easier to solve than elastic (or partially inelastic) ones, because there are fewer degrees of freedom in the final state (only one velocity, not two).

4.5.4: Partially Inelastic Collisions

As suggested by their name, a partially inelastic collision is one where some kinetic energy is lost in the collision (so it isn’t elastic) but not the maximum amount. The particles do not stick together, so there are in general two velocities that must be solved for in the “after” picture, just as there are for elastic collisions. In general, since any energy from zero (elastic) to some maximum amount (fully inelastic) can be lost during the collision, you will have to be given more information about the problem (such as the velocity of one of the particles after the collision) in order to be able to solve for the remaining information and answer questions.

4.5.5: Dimension of Scattering and Su cient Information

Given an actual force law describing a collision, one can in principle always solve the dynamical di erential equations that result from applying Newton’s Second Law to all of the masses and find their final velocities from their initial conditions and a knowledge of the interaction force(s). However, the solution of collisions involving all but the simplest interaction forces is beyond the scope of this course (and is usually quite di cult).

The reason for defining the collision types above is because they all represent kinematic (math with units) constraints that are true independent of the details of the interaction force beyond it being either conservative (elastic) or non-conservative (fully or partially inelastic). In some cases the kinematic conditions alone are su cient to solve the entire scattering problem! In others, however, one cannot obtain a final answer without knowing the details of the scattering force as well as the initial conditions, or without knowing some of the details of the final state.

To understand this, consider only elastic collisions. If the collision occurs in three dimensions, one has four equations from the kinematic relations – three independent momentum conservation equations (one for each component) plus one equation representing kinetic energy conservation. However, the outgoing particle velocities have six numbers in them – three components each. There simply aren’t enough kinematic constraints to be able to predict the final state from the initial state without knowing the interaction.

Many collisions occur in two dimensions – think about the game of pool, for example, where the cue ball “elastically” strikes the eight ball. In this case one has two momentum conservation equations and one energy conservation equation, but one needs to solve for the four components of two final velocities in two dimensions. Again we either need to know something about the velocity

102Or more properly, into Enthalpy, which is microscopic mechanical energy distributed among the atoms and molecules that make up an object.