I think it’s time for a good old practice problem. This is a pretty basic one, which you might encounter as a freshman physics major. The general concept of dealing with inelastic collisions is one that you never escape, and from special relativity to quantum mechanics this type of thing keeps appearing in new contexts. This particular problem is from an old GRE practice exam, but I think solving it is kosher since it’s not original to them and you can’t copyright facts.
In a nonrelativistic, one-dimensional collision, a particle of mass 2m collides with a particle of mass m at rest. If the particles stick together after the collision, what fraction of the initial kinetic energy is lost in the collision?
Let’s look at the concept first. Why should any energy be lost in the collision? Imagine sticking a magnet to your refrigerator. As you bring the magnet close, you’ll feel a force from the attraction of the magnet and the refrigerator surface. That force pulls the magnet in, and a force through a distance is work. The energy involved in this process is the binding energy, and you’ll have to do that much work to pull the magnet away from the refrigerator. On the level of individual particles this happens too. Particles that stick together will have a binding energy associated with their being bound. That’s not the only way colliding things can lose kinetic energy, of course especially for macroscopic objects, heat, sound, and elastic deformation are all possibilities as well.
On to the problem. The initial energy is (1/2)(2m)v^2The initial momentum by the same token is mv. After the collision, the momentum must be conserved. Let’s denote the initial velocity with a 0 and the final velocity of the two particle system with the letter f and write down the conservation of momentum.
Which after collecting terms becomes
Plug that into the kinetic energy equation and we get
Which is 2/3 of the original kinetic energy, meaning 1/3 of the original kinetic energy was lost.