# Energy loss in sticky collisions

A typical problem of sticky collisions involves an object colliding with an object at rest on frictionless surface, and the two move together. If conservation of momentum is applied we get

$$v'=\frac{m} {m+M} v$$

If conservation of energy is applied we get

$$v'=\sqrt{\frac{m} {m+M}} v$$

What I don't get is why energy is not conserved here. Is it merely because the two expressions don't match up and we assume one of them is true? Why is the first one chosen to be correct? If the first one is correct does that mean the conservation of momentum in this case implies that energy is lost? In which form is this energy dissipated?

First, you should understand what "conservation" means in a physics sense. Conservation of a quantity means it is neither created nor destroyed in a time-limited process (we're not talking about eternity past or future). It is merely moved around to other systems/objects or (for energy) changes forms. The means that if a quantity (like momentum or total energy or total charge) in a system changes, then some of the quantity is added or removed in the system. Conservation includes the possible "flow" of the quantity in or out. If there is a totally isolated system, then conservation implies constancy. If the quantity is not constant, the means the system is not isolated.

In your question, you have inferred that the system of the two objects is isolated. That means total linear momentum, total energy, total angular momentum, and total charge (all of which are always conserved) should be constant. Obviously, the momentum of $$m$$ changes (conserved but not constant) and the momentum of $$M$$ changes, but the total momentum of the two objects, if they constitute an isolated system, remains constant: $$\vec{p}_{after}=\vec{p}_{before}+\vec{J}_{external},$$ where $$\vec{J}_{external}$$ is the transfer of momentum (impulse) due to external forces (gravity, friction, etc). Since impulse is zero, the momentum remains constant.

Next, the total energy of the system is conserved, and should be constant if you discount any sound waves created by the collision, but you have looked only at the kinetic energy of the system. There is no conservation law for kinetic energy!

There are special types of collisions in which the total kinetic energy of the objects is constant, and those collisions are called perfectly elastic collisions. That is a special case and NOT a general conservation law. Unfortunately, the term "conservation of kinetic energy" has sneaked into the lexicon, but it is a misnomer.

So, finally, what happened to the kinetic energy in your collision. If two objects stick together, there must be some type of permanent deformation of interacting surfaces (or some change in field potential energy if charges or magnets are involved). Much like a spring getting compressed and never releasing, the potential energy has increased permanently if the objects are stuck together. Also, some of the kinetic energy shows up as an increase in internal energy (hit a nail with a hammer several times an notice the temperature increase of both objects). Finally, sound waves transfer energy out of the system.

• I guess conservation of K. E was ingrained in me and I didnt think much about it Thanks for clearing it up for me – ben tenyson Sep 23 at 16:15

What you are talking about is an inelastic collision, which is defined by loss of kinetic energy (converted into e.g. heat or sound, as you had before the edit). Check this wikipedia page for more info.

On a side note, if you have two expressions that tell you different things, this means that at least one of them is wrong. You don't just choose one of them to be true, and disregard the other - you have to go back and check whether all the assumptions you have made make sense.

The mistake you are making is saying the second equation is due to "conservation of energy". The equation only addresses conservation of kinetic energy. Kinetic energy is not conserved. But total energy is conserved. The kinetic energy that is lost is dissipated primarily as heat because the collision is inelastic.

Hope this helps.