# Why is kinetic energy conserved in elastic collisions and not inelastic collisions?

I realize this has been asked many times around the Internet, but I've found nothing that answers my question. I'm going to use an example situation of PAScars traveling along a PAStrack (negligible friction). For elastic collisions, they simply collide, for inelastic collisions, they collide with Velcro strips that attach, forming one mass.

I've seen that kinetic energy is lost in inelastic collisions as it transforms into other forms of energy (sound, deformation, head). How and why is this same transformation not occurring for elastic collisions? There's collision - contact - so surely there's a noise, and surely the two colliding objects deform partially, and surely there's heat produced from that collision.

I simply can't see what mystical act is happening in elastic collisions that prevents energy from transforming from kinetic energy into other forms, while, for inelastic collisions, there's seemingly nothing preventing this change from happening.

• The dynamics carts with magnetic bumper make no noise when they collide, there's probably still a bit of heat produced by eddy currents though. Oct 21, 2016 at 1:18
• "Why is kinetic energy conserved in elastic collisions" -although you might find this unsatisfactory, it is the case that this defines elastic collision. Conservation of energy and momentum are given and, in general, some fraction of KE is converted to other forms. Thus, we need a name for the (idealized) collision in which the fraction of KE converted is zero. Oct 21, 2016 at 1:22

How and why is this same transformation not occurring for elastic collisions? There's collision - contact - so surely there's a noise, and surely the two colliding objects deform partially, and surely there's heat produced from that collision.

Yes, you are right. Elastic collisions are an abstraction or an idealisation at the macroscopic scale. In other words there is no such thing as a completely elastic collision in practice, because any real collision between objects gives off noise or deforms the colliding bodies partially. However, at a microscopic scale, you can easily have elastic collisions between atoms or other small particles such as the molecules in a gas.

I simply can't see what mystical act is happening in elastic collisions that prevents energy from transforming from kinetic energy into other forms, while, for inelastic collisions, there's seemingly nothing preventing this change from happening.

It's not mystical, it's an idealisation. This form of abstraction is very common in science. For example at ordinary temperatures there is no such thing as a conductor with zero resistance, but still we teach circuit theory of inductances or capacitors with wires with zero resistance, because the resistance can be neglected. It's an idealisation of reality.

I think you're confused by the nomenclature, because "elastic" and "inelastic" aren't your only two options --- there's a middle ground where most real collisions live, which is called "partially elastic" or "partially inelastic" depending on your mood.

An elastic collision is defined as a collision where the total kinetic energy of the interacting objects is the same before and after the collision. When you have two objects that collide, you can measure or compute the total kinetic energy before and after the collision; if the energy is conserved, you say "that was an elastic collision" just because "in this collision the total kinetic energy was a constant" has too many syllables. A completely elastic collision is a special case.

A completely inelastic collision is the other extreme of the special case, where the two objects end the interaction with zero momentum relative to each other. This represents the maximum energy that can be lost, because there's a reference frame where the final state of the system is at rest. Completely inelastic collisions are relatively easy to produce by making your projectiles stick together, as you point out in your question.

Real collisions generally live in the middle. You're right that a real collision makes sounds and generates heat, and that this energy comes from the initial kinetic energy of the system. However if the energy that's lost is a small fraction of the total initial energy, then we can say that the collision was approximately elastic, which makes the analysis easier.

In quantum-mechanical systems that don't have any internal degrees of freedom, it's possible to have a completely elastic collision. But macroscopic collisions always lose some, perhaps negligible, amount of energy.

(Sometimes you even hear about "super-elastic" collisions. These are collisions where some internal energy source is converted to kinetic energy. For instance, a gun is fired by striking the ammunition so that the gunpowder ignites and the bullet is propelled from the barrel; the collision between the moving part of the trigger mechanism and the bullet at rest is super-elastic, because there's more kinetic energy after the gun is fired than before.)

• Thanks for your clarification! I definitely was mixing up conceptual idealizations and physical realities. If I may bother you more, I'm still slightly confused as to why there's such a large disparity in kinetic energy dispersed in partially inelastic versus partially elastic collisions. What about inelastic collisions makes them disperse so much more kinetic energy than elastic collisions, even when, in reality, they're both likely to suffer kinetic energy loss from similar sources? Oct 21, 2016 at 1:58
• @ThatGuy7 The difference depends on the details of the interaction. For instance, imagine a head-on collision between an automobile and an immovable building. If you were to put a bunch of bouncy rubber balls on the front of the car, so that the collision were approximately elastic, then the car would bounce backwards with $\vec p_\text{final} \approx -\vec p_\text{initial}$. But real cars are built so that the front end crumples up and the car comes to a stop in even a fairly minor collision, with $\vec p_\text{final} = 0$. That cuts the impulse $\Delta p$ on the occupants roughly in half.
– rob
Oct 21, 2016 at 2:24

In every collision energy is conserved.
If there are no external forces acting then momentum is conserved.
For some collisions kinetic energy is conserved and these are called (perfectly) elastic collisions.
In practice when a real world collision occurs it can be called elastic if very little of the kinetic energy is lost.

If you think of the bonds between the atoms in a material as little springs then as the bodies interact the kinetic energy is converted to spring potential energy.
That spring potential energy is then released as kinetic energy.
If those little springs convert all their potential energy into kinetic energy then the collision is elastic.

This video of a golf ball hitting a surface shows that a lot goes on during a collision.

Inelastic collisions still conserve energy but some of the kinetic energy can be converted into heat and sound, and the permanent deformation of materials (permanently breaking bonds - those little springs between atoms).

Remember the golf ball hitting the surface and oscillating after rebound?
That is an example of an inelastic collision because those oscillations you saw after the collision will die down due to frictional forces and the end result will be that the golf ball has got hotter. The energy has come from the kinetic energy of the golf ball.
In Squash before actually starting a match the players use the ball for a number of minutes to warm it up.

There are also super-elastic "collisions" where the kinetic energy increases with a good example being a round being shot from a gun.
Before the gun is fired there is no kinetic energy but after the gun is fired the round and the rifle both have kinetic energy.

There is no mystery. All collisions conserve momentum and energy, but the incident kinetic energy can be transmuted into other forms, such as heat or sound or elastic or plastic deformations.

It is entirely a matter of definition that we call those which preserve kinetic energy as "elastic" and all others as "inelastic". We tend to concentrate on elastic collisions, especially when learning physics, because they are easier to calculate and many collisions are approximately elastic.