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I've been trying to get a good grip on the difference between conservation laws. Momentum is particularly tricky, I don't understand how quantities like $m\mathbf v$ can be conserved when other things like deformation and heat come into play. Googling it seems to suggest that the answer lies in energy being a scalar and momentum being a vector, but then this situation still confuses me:

Two cars with the same mass and different velocities, say $-5\mathbf i$ and $10\mathbf i$, collide and no external forces act.

All their kinetic energy goes into sound, heat, and deformation. So final velocity is reduced to zero for both of them.

What happened to the momentum? Their vectors were unequal so it can't be zero, but the final velocities are zero.

Thanks for any help.

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How did physics end up with momentum and energy conservation?

By observing, experimenting and cogitating over the results of kinematic experiments the laws of conservation of energy momentum and angular momentum were discovered to hold. The whole mathematical model of Classical Mechanics, is successful and has these laws inherent in the formulation BECAUSE this is what has been observed and measured over and over again.

When building up a thought experiment , it is not wise to question basic laws in the framework where they were derived. One does not have to think "why" , about conservation laws in their appropriate frame , but "where am I going wrong in this thought experiment".

Two cars with the same mass and different velocities, say −5i and 10i, collide and no external forces act.

Let us suppose they have the same mass for simplicity, then their center of mass moves with 5i/2.

Conservation of momentum means that the mess after collision moves with 5i/2 velocity and the momentum will be 5i(2M)

Of course it finally comes to rest due mainly to frictional forces that stop the slide ( or other objects scattered in the way).

All their kinetic energy goes into sound, heat, and deformation. So final velocity is reduced to zero for both of them.

Some will go to sound heat and deformation, and part of the momentum will be transferred to atoms and finally the earth, but a lot of the 5i(2M) will remain for the mess to move after scatter and do extra damage.

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The momentum of the composite of two cars is the vector sum of the momentum of car 1 plus that of car 2. That composite momentum is the same before and after the collision. This assumes they are colliding above the ground, or they are on a frictionless surface.

Example: two 1kg sticky bean-bags are moving together. One is traveling to the right at 3m/s. The other is traveling to the left at 1m/s. The composite center of mass is 2kg traveling to the right at 2m/s. After the collision, you have a 2kg mass moving to the right at 2m/s.

On the other hand, if they are elastic they will bounce apart and be two separate masses again. But you can rely on it that their composite center of mass is just as heavy and continues in the same direction, with the same speed, as it did before the collision.

If they are on the earth's surface and there is friction, so they stop, then you also have to consider the momentum of the earth in the composite as well. Since you probably don't want to do that, that's why people make idealizing assumptions.

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  • $\begingroup$ So in the example I gave the initial momentum of the system would be 5 (whatever unit) to the right. But after if the final velocities of both objects are zero or close because of energy loss to things like heat and deformation, and this 5 can't be obtained with what is left in the system, what happened? $\endgroup$ – BoddTaxter Dec 16 '15 at 2:02
  • $\begingroup$ @AdamGraehling: You're talking about momentum, not energy, so forget about heat and deformation. Those are conservation of energy, not conservation of momentum. In your example, when you say "no external forces act", that means no friction, like if the cars are on a greased surface. If so, the cars will not stop after colliding - the pileup will continue moving at the same speed as their center of mass did before. $\endgroup$ – Mike Dunlavey Dec 16 '15 at 12:56
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Your assumption (intuition?) that the resulting lump would be not or little moving is simply wrong. The resulting lump would move with 2.5i, as we can easily calculate:

-5i * m + 10i * m = 5i*m -> that is the total momentum before and after the impact.

As the total mass of the lump after the impact is 2m, it moves with 5i*m / 2m = 2.5i.

If this is hard to imagine, then consider the case that car 1 is going 80 miles per hour, and car 2 is just rolling slowly, with only inches per second. Do you really think after the impact the result would sit still? The lump would move with just short of 40 mph in the same direction (after a nice loud bang).

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  • $\begingroup$ I understand your math and I agree, my confusion comes in reconciling it with conservation of energy. In an idealized system the initial energy of the system would be the the KE of the cars, which depends only on their mass and velocity. During the collision energy will be transformed into a couple other forms like heat, material deformation, and sound. So final energy consists of kinetic, heat, and other energies and still must equal those initial kinetic energies. So what if so much energy is changed to those other forms that the final KE means a velocity less than 5 or even zero? $\endgroup$ – BoddTaxter Dec 16 '15 at 5:26
  • $\begingroup$ that's exactly what the conservation laws say - it is not going to happen. The resulting lump will move with the calculated speed, and the energy that gets converted is exactly the difference between the KEs before and the KE afterwards. $\endgroup$ – Aganju Dec 16 '15 at 5:27
  • $\begingroup$ KE before: 1/2 * 100 i^2 * m + 1/2 * 25 i^2 * m = 62.5 i^2 * m; KE afterwards: 1/2 * 6.25 i^2 * 2m = 6.25 i^2 * m; difference = 56.25 i^2 * m which is 90% less. So exactly 90% of the KE will be converted in heat, deforming, etc., and 10% will remain in KE. The law predicts that, and the experiment will show that it is correct. (go try it, but please not with my car) $\endgroup$ – Aganju Dec 16 '15 at 5:31
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I'd imagine the momentum went in to the air molecules too, no? That and the molecules of the asphalt. Perhaps the friction is like a huge number of micro-collisions occurring, so the momentum is gradually transferred to the asphalt molecules while the cars together slow down.

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  • $\begingroup$ This is not an answer and belongs in the comment section. $\endgroup$ – Gert Dec 16 '15 at 3:33
  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. - From Review $\endgroup$ – Gert Dec 16 '15 at 3:34
  • $\begingroup$ Is it not an answer? I believe that it is the case, so how is it not an answer? $\endgroup$ – SSD Dec 16 '15 at 4:01
  • $\begingroup$ Yeah I'm pretty sure that my answer is actually correct, so I don't get why you'd say it belongs as a comment. Should I have simply treated it with more conviction? "The momentum DID go into the air and asphalt molecules" There. $\endgroup$ – SSD Dec 16 '15 at 4:04
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Yes! I think momentum can be zero in case you consider air as suggested by Mr. SSD. You can imagine the scene, as in initial case the both cars are moving and wind isn't blowing. Then two cars collide and lose their KE and momentum to air. Then a soothing warm (because of heat of collision) wind blows(to conseve the momentum) with the two cars lying at rest.

In that case you have to consider the air (the mass of air that will participate will be very difficult to figure out) as a part of your system along with those two cars.

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@Adam what do you mean by "what is left in the system". Consider this example, It will help you think: Consider an astronout floating (at rest wrt you) in space with a ball in hand. If he throws the ball in a direction. Do you know what will happen? The astronout applied force F on the ball and thus the ball also applied force F on him by using newtons 3rd law.Thus the ball and astronout if considered as a system will have zero acceleration as two internal forces cancel out to give a net zero force on system. But if only ball or only astronaut is a system then they have the Net external force F acting on them. Thus astronout and ball will have to move. The direction of astronout's motion is in oppossite direction in which ball is thrown. They move in such a way that their centre of mass is always at rest. But individually they are in motion as they had an net external force applied on them. *The scene was set in space to avoid other effects like friction etc. *The internal forces(due to third law they cancel in pairs) can't produce net force on a system and thus no force means no change in momentum and thus momentum is conserved in absense of external forces.This is conservation of momentum principle.

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