# Why was momentum not conserved

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## Main Question or Discussion Point

Elastic:
A ball of mass much less than the wall collides with the wall. If the collision is elastic the kinetic energy of ball is unchanged. It means that kinetic energy of wall is zero before and after the collision. Therefore velocity of wall HAS NOT CHANGED. When Velocity is Zero momentum is zero.
The change of momentum of ball is 2mv while the change of momentum of wall is ZERO.
Momentum is not conserved.

Inelastic: ball collides with the wall and due to the glue applied on the surface of ball the ball sticks on the wall after collision. The change of momentum is mv. The change of momentum is zero as the wall doesn't move. The kinetic energy is converted into internal energy and sound energy. Momentum is although not conserved again.

As mass of wall is much bigger the change in velocity of ball has to be small but not zero. Here it is zero.
In a video Professor walter lewin said that in this case the momentum of wall is NOT zero but kinetic energy of wall is zero. He said that it can be mathematically proven it but i don't this it is possible.
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## Answers and Replies

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The assumption that the wall doesn't move after the collision is only valid in the limit as ##\frac{m}{M}## goes to zero, where ##m## is the mass of the ball and ##M## is the mass of the wall. If you set everything up in the general case wherein you make no assumptions about ##\frac{m}{M}## and then take the limit as this dimensionless parameter goes to zero then you will see the proper behavior of the conservation laws.
The assumption that the wall doesn't move after the collision is only valid in the limit as ##\frac{m}{M}## goes to zero, where ##m## is the mass of the ball and ##M## is the mass of the wall. If you set everything up in the general case wherein you make no assumptions about ##\frac{m}{M}## and then take the limit as this dimensionless parameter goes to zero then you will see the proper behavior of the conservation laws.
There is a property of matter called inertia and it can not be ignored. Such a force can not make it move.
Professor walter lewin is correct. Lets do the elastic case. The ball momentum changes by -2mv. The momentum of the wall must change by 2mv due to momentum conservation. Assuming that the wall was initially at rest, the speed of the wall will be Vw = 2mv/Mw. It's kinetic energy will be Kw= (1/2) Mw Vw2 = (1/2) Mw (2mv/Mw)2 = 2(mv)2/Mw. Now if you take Mw → ∞, you get Kw → 0 while the momentum of the wall will be pw = 2mv ≠ 0.
arildno
For a fixed wall, you can perfectly well have an elastic collision (meaning that kinetic ENERGY is conserved), even though momentum is not conserved.
If external reaction forces holding the wall fixed are comparable in magnitude to the collision forces, there is no reason why momentum should be conserved.
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Another familiar, somewhat analogous example is a ball hitting a pendulum.
In this case, you will have conservation of angular momentum around the pivot point, but the impulse acting AT the pivot point is comparable to the collision impulse where the ball hits the pendulum. Linear momentum is certainly not conserved in this scenario, although the angular momentum is. Such a scenario can perfectly well be elastic, though.
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For a fixed wall, you can perfectly well have an elastic collision (meaning that kinetic ENERGY is conserved), even though momentum is not conserved.
If external reaction forces holding the wall fixed are comparable in magnitude to the collision forces, there is no reason why momentum should be conserved.
I think it is implied in the question that there are no external forces (A wall floating in space) but the Original poster might want to clarify that point.
arildno
I think it is implied in the question that there are no external forces (A wall floating in space) but the Original poster might want to clarify that point.
I ought to have said that for a wall floating in space, momentum might certainly well be conserved, and the reason why we may regard the velocity of the wall to be zero, is that it is many orders of magnitude less than the velocity of the ball. (The resulting momenta, though, will be of the same order, indeed perfectly equal but oppositely directed)
Professor walter lewin is correct. Lets do the elastic case. The ball momentum changes by -2mv. The momentum of the wall must change by 2mv due to momentum conservation. Assuming that the wall was initially at rest, the speed of the wall will be Vw = 2mv/Mw. It's kinetic energy will be Kw= (1/2) Mw Vw2 = (1/2) Mw (2mv/Mw)2 = 2(mv)2/Mw. Now if you take Mw → ∞, you get Kw → 0 while the momentum of the wall will be pw = 2mv ≠ 0.
This is very unintuitive. It also says the velocity of the wall Vw = 2mv/Mw goes to zero, then it would seem the momentum of the wall is finite although its velocity is zero?
The momentum is ##p_w = M_w v_w##. We are taking the limit as ##v_w## goes to zero and ##M_w## goes to infinity so the momentum remains finite. This is why I said in post #2 to write things out generally and then take limits, which is what dauto later did anyways.
The momentum is ##p_w = M_w v_w##. We are taking the limit as ##v_w## goes to zero and ##M_w## goes to infinity so the momentum remains finite. This is why I said in post #2 to write things out generally and then take limits, which is what dauto later did anyways.
I see. So for the kinetic energy, Mv2 goes to zero as M goes to inifnity and v goes to zero, because v2 goes to zero much faster than M gets large?
And strictly speaking, do we have to say we take the double limit?
I see. So for the kinetic energy, Mv2 goes to zero as M goes to inifnity and v goes to zero, because v2 goes to zero much faster than M gets large?
That's right. The energy is proportional to the square of the speed, so it is an "infinitesimal" of higher order.
This is very unintuitive.
Infinities are. When you keep M constant, and let m -> 0, the effect on m/M is the same, but you don't get the mismatch between momentum and energy limits.
Elastic:
A ball of mass much less than the wall collides with the wall. If the collision is elastic the kinetic energy of ball is unchanged. It means that kinetic energy of wall is zero before and after the collision. Therefore velocity of wall HAS NOT CHANGED. When Velocity is Zero momentum is zero.
The change of momentum of ball is 2mv while the change of momentum of wall is ZERO.
Momentum is not conserved.

Inelastic: ball collides with the wall and due to the glue applied on the surface of ball the ball sticks on the wall after collision. The change of momentum is mv. The change of momentum is zero as the wall doesn't move. The kinetic energy is converted into internal energy and sound energy. Momentum is although not conserved again.

As mass of wall is much bigger the change in velocity of ball has to be small but not zero. Here it is zero.
In a video Professor walter lewin said that in this case the momentum of wall is NOT zero but kinetic energy of wall is zero. He said that it can be mathematically proven it but i don't this it is possible.
Momentum is conserved if the laws of physics are invariant under spatial translations (i.e. if they are the same everywhere in space). If you have an immovable wall somewhere, the laws of physics are NOT invariant under spatial translations, because there is a law that says that when you arrive at the wall something happens (you elastically rebound, or stick to it, or whatever).

So why would you expect momentum to be conserved when you treat the wall as immovable? It's not - you're right. (Of course in the real world there's no such thing as an immovable wall, but you're not asking about the real world.)
This ball-wall problem is worked in detail here:
http://www.lhup.edu/~dsimanek/ideas/bounce.htm

Momentum is conserved in the absence of external forces.

Both cases are discussed here: