Conservation of momentum | Physics | Khan Academy

When we shoot a cannon, not only does the ball go forward, but the cannon itself goes backward. This means when we shot it, the ball gained a forward momentum, and the cannon itself also gained a backward momentum. The big question is: if we know what the forward momentum of the ball is, can we figure out what the backward momentum of the cannon is? Because that's important when you're designing this, right? For safety and everything. So how do you figure out the cannon's backward momentum?

Well, we're going to figure out how to do that by one of the most powerful principles of physics—that is conservation of momentum. So let's begin. To figure out its momentum, the first question that comes to my mind is: is there an equation that connects, I don't know, maybe forces and momentum? And yes, the answer is yes. We've seen it in a previous video. We saw that the net force acting on an object equals the rate of change of momentum of that object.

Now I could try to use this, but there's a problem. I don't know the net force acting on my cannon. I don't know how much force is acting during when we fired it. I also don't know for how long that force was acting. So I don't know if I can use this equation. But wait a second, let's think about the forces acting over here a little bit more carefully. Who is pushing on this cannon? I mean, somebody must be pushing back on this cannon, right? Otherwise, it would have moved backwards. Who is doing that?

Remember, cannons or stuff cannot push on themselves. Objects cannot exert a force on themselves. All different kinds of forces that you can think of or interactions that you can think of must be between two particles or two objects. Like in this particular case, I can actually see the two objects over here: the cannon and the ball. So if I call this ball A, then we know it's the cannon that is pushing the ball forward. So it is pushing, it’s putting a force forward on A. The cannon is the one that’s pushing it forward.

Therefore, from Newton's third law, we know that the ball must be pushing the cannon backward. So it's the ball that is pushing, it's putting a force on the cannon. Let's call the cannon our object B over here. Now there will be other forces acting over here, like friction and air resistance and stuff, but let's ignore them. Let's assume that those forces are very negligible. So we'll assume that those forces are not there.

Okay, then these are the only two forces acting over here. Then let's see what happens if we apply this equation to these two objects. So let's do that. If I apply that equation for our object A, which is the ball, we can say the net force, the total force acting on A, which is basically this one force because there’s only one force over here acting on A, should equal the rate of change of momentum of A.

Okay, and similarly, I can now write the total force or the net force acting on B, which is this force, because there are no other forces. That’s what we are assuming—that should be the rate of change of momentum of B, of the cannon.

Okay, but remember from Newton's third law, these two forces are equal and opposite because we're assuming that these are the only two forces acting. There are no other forces, so these two must be equal and opposite. In other words, this must be equal and opposite to this. So let's write that down, and let’s write that condition.

If A and B are only interacting with each other—and when I say interacting, I mean putting forces, they’re putting forces only on each other, no other forces, I mean—then these two forces must be equal and opposite. Therefore, this must be equal and opposite to this one. So let's write that down. The rate of change of momentum of A must be equal and opposite to the rate of change of momentum of B.

And now, what about this delta T? What about this time? Well, this time represents the time over which the force was acting on A, and this represents the time over which the force was acting on B. But look, for whatever time the cannon is pushing the ball, for the same time the ball will be pushing back on the cannon, right?

I mean, if the cannon was pushing on the ball, say, I don’t know, maybe for a couple of milliseconds, the ball will also be pushing back on the cannon only for a couple of milliseconds. Once the contact is broken, then they're no longer putting a force on each other. Therefore, since these are the two equal and opposite forces, they must be acting on each other for exactly the same time. This time must be the same as this one, so I can cancel this out.

What I find now is that the change in momentum of A should exactly equal, but opposite to, the change in momentum of B. And this—we're going to box this along with the condition—because that condition is going to be super important for us.

Okay, so what is this telling us? This is saying that whatever momentum object A gains, object B will lose. I say lose because there’s an opposite sign. So if this gains, B will lose, but whatever this gains, it'll lose the same amount of momentum.

So whatever this gains, this will lose the same amount of momentum. In other words, the total momentum of both the cannon and the ball together does not change after firing it. The total momentum stays the same. This is called the principle of conservation of momentum.

A helpful analogy over here to think about would be in terms of money. So imagine I had some $100 with me and you had some $200 with you. Okay, now if there's a condition which says that we are only going to be exchanging money amongst ourselves, okay, then notice whatever money I lose—if I lose some $20, you would have gained that $20 because you’re only exchanging money amongst ourselves. Whatever money you lose—if you lose, I don’t know, maybe $50—I would have gained that $50 because we are only exchanging money amongst ourselves.

So whatever amount of money we exchange amongst ourselves, the money that I have will change, the money that you have will change, but our total money would stay the same: $300 because we're only exchanging money amongst ourselves. That's what's happening over here.

Because these two objects are only interacting with each other, whatever momentum one gains, they’re exchanging momentum over here. Whatever momentum one gains, the other one must be losing, therefore the total momentum that they have stays the same.

But now, you can also understand and appreciate why this condition is important. I mean, if you come back over here, if we started exchanging money with a third person, if we started exchanging with a third person, then this will not work. Then the total money that you and I have will change because now we’re now there’s a third person involved over here.

That's the reason why this only works as long as the two objects are only exerting forces interacting with each other, exchanging momentum with each other, or putting forces on each other. If there are forces from outside these two objects, we can call this as our system, and if there are forces outside of the system—I mean unbalanced forces from outside the system—balanced forces outside the system will not affect us, but if there are unbalanced forces outside the system, then no, the momentum of this system will not be conserved anymore.

And the beauty is this can be extended to any number of objects. Imagine now if you had three objects, and again under the condition that they're only interacting with each other, then the total momentum of those three objects should stay the same. Again, the condition is that there shouldn’t be a fourth object that they are interacting with. Then that will be an external force; that will change the momentum of those three objects together. That makes sense, right?

This is the powerful conservation of momentum. Alright, now let's see if we can apply this to answer our original question. So we'll consider the case before we shot the cannon. So remember, momentum is mass times velocity. Right now, before we shot the cannon, the cannon was at rest, so its velocity was zero. Its momentum was zero. The ball was also at rest; its velocity is zero, so its momentum was also zero.

So the total initial momentum before firing was zero, which means after firing—since we’re assuming that they’re only interacting with each other—the total final momentum must stay the same. That's what the principle of conservation of momentum says—that the total final momentum must equal the total initial momentum.

So the total final momentum must also be zero. Now, we might be thinking, well, how can that be? How can the total final momentum be zero? Because they are now moving, so there is some momentum. Well, remember, momentum is a vector quantity. So the only way the total final momentum can be zero is if whatever is the forward momentum that this ball has, this cannon must have the exact same backward momentum.

So whatever forward momentum the ball must be having, the cannon must be having the exact same backward momentum because when that happens, it's a vector; they can now cancel out because they’re in the opposite direction. That's the only way they can conserve momentum, which means we've answered our original question. The backward momentum of the cannon must be exactly the same as the forward momentum of that ball.

And since we know how to calculate the momentum—since we know momentum is mass times velocity—we can now also understand that since this has much more mass than this, they have the same momentum. This is more massive; they should have less velocity. And so it makes sense that the ball travels a lot faster compared to how much, you know, the cannon travels backwards.

So they'll have different speeds, but they should have the same momentum. But this now brings us to the final part, which is you may be saying, well, yeah, but remember in reality we do have frictional forces over here. We do have a resistance over here, right? So technically, technically speaking, in reality, these are not the only two forces acting. There are other forces acting.

So there are external forces on my system. My system is the cannon and the ball together. So technically, momentum is not conserved because this condition is violated. There are external forces, right? So technically speaking, this is not true, right? And I'll say, yes, technically, that is not true. But here's the beauty—remember we are considering that initial and the final, when we're talking about, we're talking about just before shooting the ball and just after that shooting the ball.

That time interval is extremely tiny. This is the time interval you’re talking about. It's a very small time interval. During that time interval, the effects of friction and the effects of air resistance are very minimal. The dominant forces are these two. These are the ones that have the maximum effect.

So even though it's true that technically these forces exist because their effects in reality—in practical actual cases—are very tiny during that small time, we can still ignore them and get a very good approximate answer. But if you take a big gap between the initial and the final momentum, if you wait for a long time after firing, then yes, momentum will definitely not be conserved because now the effects of friction and air resistance will be definitely felt.

You will see that this cannon will eventually come to a stop. You will see this ball would then fall into the ground and eventually come to the stop. Now, clearly, the momentum has gone out of the system. Now the effects of other forces cannot be neglected.

But the key idea is if you use it for a very short time, while that explosion happened—during which that explosion happened—then we can ignore the effects of all the other forces because these will be the dominant forces, and we will still get a pretty, pretty good approximation.

Which means conservation of momentum is a powerful tool to use when you're dealing with forces that last for small times, like explosion forces, or when we're dealing with collisions. Whenever such things happen, we’re going to use conservation of momentum. It's going to be our go-to tool.