Topic 4.3 gave you equations that hold for every collision: one in a line, two in a plane. They are often not enough. Two carts that collide in a line have two velocities afterward, and one equation cannot find two unknowns. This topic supplies the missing piece, and it is a statement about energy rather than momentum.

1. Elastic means the kinetic energy survives

A collision is elastic when the total kinetic energy of the system after the collision equals the total kinetic energy before it.

\(\sum K_{\text{before}} = \sum K_{\text{after}}\)elastic, and only elastic

Read that carefully, because it is a statement about the total. The kinetic energies of the individual objects are free to change, and in almost every elastic collision they do. One cart can leave with much more than it arrived with, as long as the other leaves with correspondingly less.

The practical value is that this is a second equation. In one dimension you now have two equations for two unknown final velocities. In two dimensions you have three for the three unknowns that remain once one angle is given. A problem that says elastic is handing you that equation, and that is what the word is doing there.

2. Inelastic means some of it is gone

A collision is inelastic when the total kinetic energy of the system decreases. That covers almost everything that happens in a real room.

The energy is not destroyed. Nonconservative forces during the contact transform it into forms that are not kinetic energy of the objects: the permanent deformation of a bumper, the sound you hear, and a very small rise in temperature of both objects. None of it comes back, which is exactly what makes those forces nonconservative.

3. Perfectly inelastic, where the most is lost

A collision is perfectly inelastic when the objects stick together and move off with the same velocity. That single sentence supplies the missing information just as cleanly as the elastic condition, because it leaves one unknown velocity instead of two.

\(m_1\vec{v}_1 + m_2\vec{v}_2 = (m_1 + m_2)\vec{v}_f\)one velocity afterward

Of all the collisions that start from the same two objects moving in the same two ways, the perfectly inelastic one loses the most kinetic energy. Nothing can lose more, because the objects would then have to pass through each other.

The reason is easiest to see through the other form of the kinetic energy. Whatever happens, the total momentum afterward is fixed. Writing \(K = p^{2}/2m\) for the system, the largest possible combined mass carrying that fixed momentum is the one where everything moves together, and the largest denominator gives the smallest kinetic energy.

4. Counting equations before you start

This unit is mostly bookkeeping once you have the ideas, and the bookkeeping is worth doing out loud. Count the unknowns, then count the equations, and only then start writing algebra.

In one dimension. Two unknown final velocities. Conservation of momentum gives one equation. If the problem says the objects stick together, that is the second. If it says elastic, the kinetic energy statement is the second. If it says neither, you will be given one of the final velocities.

In two dimensions. Four unknowns if nothing is given: two speeds and two angles. The component equations give two. Elastic gives a third. The fourth has to come from the problem, which is why these questions always hand you one angle or one speed.