Conservation of momentum and conservation of energy

Question

I had the impression that conservation of momentum is valid only when we consider no loss of kinetic energy into heat etc. My tutor says that even if we have loss of kinetic energy as heat in a collision, we still have conservation of momentum. Is this true?

EDIT: Alright Bill, but tell me something please: We have two cars that collide and stay together. Or two balls of plasteline or mud or whatever. Obviously all the momentum of the system is lost. How do you say that always momentum is conserved? For this momentum conservation theorem to work, do we ASSUME that kinetic energy is conserved? Does it ONLY work for perfect elastic collisions? Please answer my specific question, because so far all the answers are about everything apart from what I am specifically asking! My specific question (in addition to the above) is: If in a collision there is a coefficient of restitution BELOW 1, doesn't that mean that the collision is INELASTIC? YES OR NO! And if that means that the collision IS INELASTIC, is it correct to use AT THE SAME TIME momentum conservation equation? YES OR NO!

Answer 1

Momentum, energy, angular momentum, and charge are conserved locally, globally, and universally. One must remember that conservation locally (within a defined system) does not mean constancy. Constancy occurs only when the system is closed/isolated from the rest of the universe.

Conservation means that these quantities cannot spontaneously change. Let's consider momentum: the momentum of a system at a later time must equal the momentum at an earlier time plus the sum of the impulses applied to a system. The impluses in this sum could be adding or removing momentum from the system, but never creating nor destroying momentum: $$\vec{p}_{later}=\vec{p}_{before}+\Sigma\vec{J}_{during}.$$

For an isolated collision, without outside influence, $\vec{J}_{during}=0$, and $\vec{p}_{later}=\vec{p}_{before}$.

For the energy: $E_{later}=E_{before}+W+Q+\mathrm{radiation}$

For angular momentum: $\vec{L}_{later}=\vec{L}_{before}+\Sigma\vec{\Gamma}_{outside}$ ($\Gamma$ is torque on system)

For charge: $ Q_{later}=Q_{before}+\int I\;\mathrm{d}t$

In the case of kinetic energy, it is not universally conserved. It can appear and disappear as energy is transformed to different manifestations:heat internal energy, gravitational, electromagnetic, nuclear, all of which are energy. The total energy is conserved in a system (not necessarily constant), with the transfer agent being work/radiation/heat. The elastic collision is defined to be one in which the kinetic energy of the system remains constant.

Note that if you define a single object as the system of interest, neither the momentum nor the kinetic energy will remain constant during a collision with another object or while it falls in a gravitational field, but the momentum will be conserved (the object is subjected to impluses) and the energy of the object is conserved (outside forces do work).

Bottom line: Define a system, look for transfers of momentum (impulse), energy (work, etc), angular momentum (torque), and charge (current) into or out of the system. Then see if any of those conserved properties are also constant for your situation.

EDIT - Response to OP specific questions:

My specific question (in addition to the above) is: If in a collision there is a coefficient of restitution BELOW 1, doesn't that mean that the collision is INELASTIC? YES OR NO!

Yes. One may also call it partially elastic. If the coefficient of restitution is zero (0), the collision is completely inelastic.

And if that means that the collision IS INELASTIC, is it correct to use AT THE SAME TIME momentum conservation equation? YES OR NO!

Yes. Momentum is conserved in all collisions and explosions. And sometimes it might even be constant for short periods of time.

Answer 2

In a collision momentum is conserved if there are no external forces. Energy is also conserved but in the examples kinetic energy is an important parameter. Elastic collisions are ones in which kinetic energy is conserved. In non-elastic or inelastic collisions kinetic energy is not conserved and some kinetic energy can be converted into heat, sound and in doing work to permanently deform the colliding objects.

You can also have super-elastic collisions where the kinetic energy actually increases (and momentum is conserved). An example is a bullet being shot out of a rifle where the kinetic energy comes from the chemical energy of the propellant.

Answer 3

I had the impression that conservation of momentum is valid only when we consider no loss of kinetic energy into heat etc.

Nope.

My tutor says that even if we have loss of kinetic energy as heat in a collision, we still have conservation of momentum. Is this true?

Yup.

EDIT: Alright Bill, but tell me something please: We have two cars that collide and stay together.

P = mv

But that P is all of the mv's in the system. So...

Lets say, to keep the math simple, that both cars mass Mc and are travelling at speed Vc towards each other. So then in that case, before the collision you have:

P = (McVc) + (MC(-Vc)) = 0

and after, it's:

P = (Mc0) + (Mc0) = 0

Momentum is conserved, it was zero before and zero after.

Wait, you cry, where did that negative sign come from? Well, what's the definition of velocity, as opposed to speed?

Mystery solved!

As always, it's in the definitions where you have to be careful. Feel free to apply it to your other examples - squishy balls and such. They'll all be the same.