# Why - momentum must be in the wall?

• PhysicsDaoist
In summary, in this lecture on elastic collisions, Professor Walter Lewin demonstrates that when a tennis ball with mass m and speed v hits a wall, it bounces back with the same speed v. The wall, having an infinitely large mass, experiences a change in momentum equal to 2mv but no change in kinetic energy. This is because momentum must be conserved in a collision, even if one object has an infinitely large mass. The change in momentum is equal to twice the original momentum of the ball, with one sign being reversed due to the collision. This can be mathematically represented using the equations for conservation of energy and momentum.
PhysicsDaoist
I was watching MIT Classical Mechanics Lecture16. On completely elastic collision, (at around 24min), professor Walter Lewin shown that when a tennis ball with mass m and speed v hitting the wall, it bouced back with speed v.
(Cut from the lecture)
<<
...Kinetic energy is conserved.
All the kinetic energy is in the tennis ball; nothing is in the wall.
The wall has an infinitely large mass, but the momentum of this tennis ball has changed by an amount 2 mv.

That momentum must be in the wall--it's nonnegotiable, because momentum must be conserved.

So now here you see in front of your eyes a case that the wall has momentum, but it has no kinetic energy.

That the wall has momentum 2 mv, it's nonnegotiable.
It must have momentum, and yet it has no kinetic energy...
>>

How can that be? The wall is NOT moving, how can it has momentum?
I velocity for wall is zero, there is NO kinetic energy nor momentum, correct? But then why did he say that? Was it a trick question?

v = p/m. If m is infinite, and p is finite, what is v?

(If you object to "infinite", recognize that the wall is attached to the Earth and plug the Earth's mass in - what is v? Would you notice a change of velocity of that magnitude?)

PhysicsDaoist said:
<<
...Kinetic energy is conserved.
All the kinetic energy is in the tennis ball; nothing is in the wall.
The wall has an infinitely large mass, but the momentum of this tennis ball has changed by an amount 2 mv.

That momentum must be in the wall--it's nonnegotiable, because momentum must be conserved.

So now here you see in front of your eyes a case that the wall has momentum, but it has no kinetic energy.

That the wall has momentum 2 mv, it's nonnegotiable.
It must have momentum, and yet it has no kinetic energy...
>>
I'm really confused by this. The lecturer said the ball bounces from the wall with twice the momentum it had when it hit the wall?

zoobyshoe said:
I'm really confused by this. The lecturer said the ball bounces from the wall with twice the momentum it had when it hit the wall?
No, its change in momentum equals twice its original momentum.

Original momentum of ball: +mv
Final momentum of ball: -mv
Change in momentum of ball: -2mv
Change in momentum of wall: +2mv

Your feet are rigid with the ground, and the ground is rigid with the wall.
When you throw the ball, it's got +mv, you incur -mv for your troubles.
When the ball hits the wall and stops, the ball looses mv and is at rest, and the wall/floor/you system gains back it's lost mv.
But the ball is compressed so it recoils back with -mv, giving the wall a push for +mv.
When you finally catch the ball, it relinquishes it's mv momentum into you, setting us right back where we started.

And if we were to be fussy about this which is probably necessary to clear up any confusion there are two main points which are relevant
1.Even ignoring gravity air resistance and so on the ball cannot bounce back with velocity v.That would be an example of a perfectly elastic collision and such collisions do not occur with macroscopic objects(they can and do with atomic scale objects for example gas atoms provided that the energy of collision is not too high)
2.The wall and everything it is attached to does move in order to conserve momentum and therefore it does pick up kinetic energy.Because of the difference in mass the energy may be vanishingly small but it is not zero and can only become zero if the wall etc had an infinite mass which it doesnt.

Doc Al said:
No, its change in momentum equals twice its original momentum.

Original momentum of ball: +mv
Final momentum of ball: -mv
Change in momentum of ball: -2mv
Change in momentum of wall: +2mv
Thanks, Doc Al. I think I get it: since the original momentum of the ball was +mv the original momentum of the wall was automatically -mv. The collision causes both signs to be reversed, and the difference between the initial and final is 2. The reason we reason that the ball has undergone a change in momentum of minus 2mv is that, to go from +mv to 0mv an equal and opposite momentum of -mv had to be given to it, and then to go from 0mv to -mv another equal and opposit momentum of -mv had to be applied. Therefore the change in momentum is -2mv. It's all about the vectors. Please correct me if I screwed that up somewhere.

zoobyshoe said:
Thanks, Doc Al. I think I get it: since the original momentum of the ball was +mv the original momentum of the wall was automatically -mv. The collision causes both signs to be reversed, and the difference between the initial and final is 2. The reason we reason that the ball has undergone a change in momentum of minus 2mv is that, to go from +mv to 0mv an equal and opposite momentum of -mv had to be given to it, and then to go from 0mv to -mv another equal and opposit momentum of -mv had to be applied. Therefore the change in momentum is -2mv. It's all about the vectors. Please correct me if I screwed that up somewhere.
You have it exactly right.

Thank you for the replies.
I think mathematically, I can work out the answer.
Simply from the fact that it is an elastic collision (assuming zero energy loss) And using conservation of energy and conservation of momentum.

v1' = (m1-m2)/(m1+m2)v1
v2' = 2 m1/(m1+m2)v1
and if m2 approaches infinity, v1' = -v1 and v2'=0

I even played with assuming m2 is 100x m1 to see how these terms vary.

However, it is interesting and difficult concept to grasp that the change in momentum absorbed by the wall is 2m1v1 when the wall velocity (v2') is zero. That is, it has zero final kinetic energy but it gain 2m1v1 momentum.

Those of you who are just starting Walter Lewin's online lectures are in for an amazing experience. Don't stop at 8.01.

Doc Al said:
You have it exactly right.

Aha! Great! Thanks much!

I need to add a few extra points of clarification.
1.An elastic collision is one where there is no loss of kinetic energy not one where there is no loss of energy.Energy is always conserved.
2.The collision between the ball and the wall is not elastic.If you could make a ball that bounced elastically with the floor you could drop it from a fixed height and it would bounce forever.
3.The wall etc does move and does get a bit of kinetic energy.
The professor should have made it clear that he was making simplifying assumptions and I am guessing that he was going on to derive the basic ideal gas equation where it can be assumed that at low enough temperatures gas molecules do indeed collide elastically.

I need to add a few extra points of clarification.
1.An elastic collision is one where there is no loss of kinetic energy not one where there is no loss of energy.

Correction: there is no loss of kinetic energy though the statement that there is no loss of energy is also true just not relative.

TheWonderer1
Read post number 9 and you will see why my reference to energy is relevant.

...
2.The wall and everything it is attached to does move in order to conserve momentum and therefore it does pick up kinetic energy.Because of the difference in mass the energy may be vanishingly small but it is not zero and can only become zero if the wall etc had an infinite mass which it doesnt.

Thank you Dadface. That is clear, my issue was with the wall and I now see where and why I was confused.

I need to add a few extra points of clarification.
1.An elastic collision is one where there is no loss of kinetic energy not one where there is no loss of energy.Energy is always conserved.
2...
3.The wall etc does move and does get a bit of kinetic energy.
...

This another clarification/correction you wrote made me realized that I put down "zero energy loss" on post 9. You are absolutely correct. I can visualize the detail much better now.

Thanks again and thank you all.

The main issue I have with the original problem statement is that the wall has infinite mass. This is createing all sort of problems, for example what is the momentum of an object with infinite mass?

Change this to a wall with a very large but finite amount of mass and everything works. You could have the wall attached to the earth, and then treat the path of the ball as a sub orbit and realize that angular momentum of ball, wall, and Earth are preserved despite any energy losses in the collision.

zoobyshoe said:
Thanks, Doc Al. I think I get it: since the original momentum of the ball was +mv the original momentum of the wall was automatically -mv. The collision causes both signs to be reversed, and the difference between the initial and final is 2. The reason we reason that the ball has undergone a change in momentum of minus 2mv is that, to go from +mv to 0mv an equal and opposite momentum of -mv had to be given to it, and then to go from 0mv to -mv another equal and opposit momentum of -mv had to be applied. Therefore the change in momentum is -2mv. It's all about the vectors. Please correct me if I screwed that up somewhere.

The original momentum of wall isn't -mv . From an inertial frame that records the velocity of the ball to be v, the wall just rests like a puppy with zero velocity as per the given statements by walter. So zero momentum.

sganesh88 said:
The original momentum of wall isn't -mv . From an inertial frame that records the velocity of the ball to be v, the wall just rests like a puppy with zero velocity as per the given statements by walter. So zero momentum.
I see your point. "Original" is certainly not the right way to put what I meant. Which was: that during the collision the wall expresses a momentum equal to the balls, opposite in direction to that of the ball. My reasoning was that the wall only manifests momentum in direct response to being impinged upon by the ball. However, I will let Doc Al sort this out.

I agree with Jeff Reid.Infinities,often accompanied by zeros,pop up in many areas of physics and if we were to take these literally it leads to contradictions and conceptual difficulties.Usually if we regard something as being infinite or, zero,we are doing nothing more than making simplifying assumptions.As an example when the ball hits the wall we can to an excellent approximation ignore the velocity change and kinetic energy etc of the wall.

zoobyshoe said:
I see your point. "Original" is certainly not the right way to put what I meant. Which was: that during the collision the wall expresses a momentum equal to the balls, opposite in direction to that of the ball. My reasoning was that the wall only manifests momentum in direct response to being impinged upon by the ball. However, I will let Doc Al sort this out.
You did garble it up a bit when you said that the original momentum of the wall was "-mv", which I should have corrected. (Sorry about that.) The key point, which I think you understand, is that the net change in momentum of the "ball + wall" due to the collision is zero. Thus the change in the ball's momentum must be equal and opposite to the change in the wall's momentum.

Let's assume, as sganesh88 stated, that the initial speed and momentum of the wall is exactly zero. Recapping what I said earlier, this time adding the initial and final momenta of the wall:

Original momentum of ball: +mv
Final momentum of ball: -mv
Original momentum of wall: 0
Final momentum of wall: +2mv
Change in momentum of ball: -2mv
Change in momentum of wall: +2mv

Make sense?

Last edited:
Doc Al said:
You did garble it up a bit when you said that the original momentum of the wall was "-mv", which I should have corrected. (Sorry about that.) The key point, which I think you understand, is that the net change in momentum of the "ball + wall" due to the collision is zero. Thus the change in the ball's momentum must be equal and opposite to the change in the wall's momentum.

Let's assume, as sganesh88 stated, that the initial speed and momentum of the wall is exactly zero. Recapping what I said earlier, this time adding the initial and final momenta of the wall:

Original momentum of ball: +mv
Final momentum of ball: -mv
Original momentum of wall: 0
Final momentum of wall: +2mv
Change in momentum of ball: -2mv
Change in momentum of wall: +2mv

Make sense?

I think so. The rules seem to be that in all cases the CHANGE in momentum = 0, but the total initial momentum of the system stays the same, albeit, redistributed. I notice that the final momentum of the wall, +2mv plus the final momentum of the ball, -mv, equals +mv, which was the initial momentum of the whole system.

zoobyshoe said:
I think so. The rules seem to be that in all cases the CHANGE in momentum = 0, but the total initial momentum of the system stays the same, albeit, redistributed.
Right. For a system to have a change in momentum, an external force is required. Considering the ball + wall as a system, their interaction is internal and cannot change the system's momentum.
I notice that the final momentum of the wall, +2mv plus the final momentum of the ball, -mv, equals +mv, which was the initial momentum of the whole system.
Right again.

Now I'm digging up in a really old thread here, but I want to clear something out. People here seem to agree that the ball must have a momentum 2mv after the collision, which I also agree on since the change in momentum must be zero.. But doesn't this imply mathematically that the ball MUST have kinetic energy? If the wall has momentum, it must must have some velocity and again then it must have some kinetic energy?

To me it seems that the the two laws of conservation of kinetic energy in an elastic collision and conservation of momentum seems to be colliding themselves?

The conservation of KE only kicks in for elastic collisions. The energy content has a lot more ways to be lost, such as through heat, sound, etc... , making them inelastic collisions, which does not conserve KE, but still conserve momentum.

Zz.

Hello center o bass.The point you are making is correct,as a result of the collision the wall picks up some momentum and some KE but since the wall is several is orders of magnitude more massive than the ball the resulting KE gain of the wall has been considered as negligible.

Last edited:
ZapperZ said:
The conservation of KE only kicks in for elastic collisions. The energy content has a lot more ways to be lost, such as through heat, sound, etc... , making them inelastic collisions, which does not conserve KE, but still conserve momentum.

Zz.

To expand upon this,to assume that the ball bounces off with an equal speed in the opposite direction is to assume that the collision is elastic which it isn't.Perfectly elastic collisions cannot happen with macroscopic objects like balls.I haven't re read the rest of the thread but it seems that basically simplifying assumptions have been made.

center o bass said:
To me it seems that the the two laws of conservation of kinetic energy in an elastic collision and conservation of momentum seems to be colliding themselves?
No, those two laws are completely compatible with each other. Whenever you get a contradiction like this the first thing to do is to look at the assumptions. In this case, an infinitely massive wall is assumed. That should be a big red flag. So, instead assume a wall with some finite mass and work the problem. Then take the limit as the wall's mass increases infinitely and you will see how it all comes out correctly.

But Zappers. It was assumed that the collision was totaly elastic, thus the ball get back all it's velocity just in the opposite direction.

So we agree that the wall must have _some_ kinetic energy, but that in the limit this goes to zero because the mass of the wall is infinite?

center o bass said:
So we agree that the wall must have _some_ kinetic energy, but that in the limit this goes to zero because the mass of the wall is infinite?

In limit its SPEED goes to zero, kinetic energy being what it is.

I think it's probably worth considering what happens when the wall has only a modest mass - say 999 times that of the ball.
The ball will rebound (elastic colllision) with -0.999 of its original velocity and the wall will move off at 0.001 of that velocity.
The energy, being kinetic, is given by mv2/2, which means that the ball will have about .999 of the energy and the wall will have only 0.001. As the mass of the wall increases, it gets less and less of a share of the original energy. The same thing applies with bullets and guns and rocket engines.
It's never going to be zero (in a classical world) but, when attached to the Earth it's going to be as near to zero as dammit.
I don't understand why this is treated as a mysterious phenomenon; the sums are pretty clear. It only gets mysterious if you start getting into the quantum aspects. Momentum must still be conserved (? afaik) but will the KE be shared in the same way as for the classical case?

## 1. Why does momentum have to be conserved in collisions?

Momentum is a fundamental concept in physics that describes the quantity of motion an object has. In collisions, the total momentum of the system must remain constant, meaning that the initial momentum of the objects before the collision must be equal to the final momentum after the collision. This is due to the law of conservation of momentum, which states that momentum cannot be created or destroyed, only transferred between objects.

## 2. How does momentum transfer between objects in a collision?

Momentum is transferred between objects in a collision through the exchange of forces. When two objects collide, they exert equal and opposite forces on each other, resulting in a transfer of momentum from one object to the other. This transfer of momentum can cause one or both objects to change their velocity or direction of motion.

## 3. Why is momentum important in understanding the behavior of objects?

Momentum is important in understanding the behavior of objects because it helps us predict how objects will move and interact with each other. By knowing the momentum of an object, we can determine how much force is needed to change its motion, and we can also predict the outcome of collisions between objects.

## 4. Can momentum be negative?

Yes, momentum can be negative. Momentum is a vector quantity, meaning it has both magnitude and direction. If an object is moving in the opposite direction of its defined positive direction, its momentum will be negative. This is often the case in collisions, where one object may have a positive momentum and the other a negative momentum.

## 5. Why must momentum be conserved in collisions with walls?

Momentum must be conserved in collisions with walls because the wall exerts an equal and opposite force on the object, causing a transfer of momentum. If momentum was not conserved, the object would either bounce back with a different velocity or continue moving through the wall, which violates the law of conservation of momentum.

• Mechanics
Replies
2
Views
894
• Mechanics
Replies
53
Views
3K
• Mechanics
Replies
3
Views
1K
• Mechanics
Replies
30
Views
2K
• Mechanics
Replies
11
Views
1K
• Mechanics
Replies
25
Views
12K
• Mechanics
Replies
22
Views
3K
• Mechanics
Replies
2
Views
2K
• Mechanics
Replies
6
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
763