### Video Transcript

In this video, we’re going to be discussing the principle of conservation of momentum. So, let’s begin by recalling what momentum is first of all.

If we’ve got an object — let’s say we’ve got this ball here — and it has a mass 𝑚. And this ball also happens to be moving — let’s say toward the right — with a velocity 𝑣. Then we can recall that the momentum of the ball, which we will call 𝑝, is equal to the mass of the ball 𝑚 multiplied by its velocity 𝑣. Therefore, any object which has mass will have momentum if it’s moving. And that momentum is equal to the product between its mass and velocity.

So, now that we’ve recalled what we mean by the momentum of an object, let’s think about the momentum of two objects. Let’s think about two different balls now, a blue one and a pink one. Let’s say that the blue ball has a mass of one kilogram. And let’s imagine that this is true for the pink ball as well; it also has a mass of one kilogram. Now, let’s also think about what would happen if our blue ball is moving toward the right, toward the pink ball, at a velocity of one meter per second. And the pink ball, in this case, is stationary; it’s not moving or, in other words, has a velocity of zero meters per second.

So, in this case, we can clearly see that because the blue ball is moving toward the pink ball and the pink ball is not moving, the blue ball is going to eventually collide with the pink ball, which is exactly what happens. But the important question is “What happens next?” Well, this is where the principle of conservation of momentum comes into the picture. This principle tells us that the total momentum of an isolated system is conserved.

Let’s break that sentence down. The easiest way to do this is to, first of all, understand what we mean by our system. Now, in physics, a system is the part of the universe that we happen to be considering. Which sounds a bit grand, but all that means is that, in this particular case, our system consists of the two balls because that’s all we happen to be considering, the blue ball and the pink ball, and crucially nothing else. Anything that isn’t the blue ball or the pink ball is said to be outside our system, which will become important in a minute. So, that is what we mean by our system.

Next, let’s understand what we mean by the total momentum of our system. To understand this, it’s probably worth rewinding a bit to before the two balls had collided. Remember, we said earlier that our blue ball was moving toward the pink ball with a velocity of one meter per second before it collided with the pink ball. Well, at this point in time, specifically before the collision, we can calculate what’s known as the total momentum of our system.

Let’s recall that the momentum 𝑝 of an object is equal to the mass of that object multiplied by its velocity. Well, the total momentum of our system is simply the momentum of all of the things in our system added together. In this case, we firstly got our blue ball, which we can see has a momentum 𝑝 subscript 𝑏 for the momentum of the blue ball. And this is equal to the mass of the blue ball, which is one kilogram, multiplied by its velocity, which is one meter per second. When we evaluate the right-hand side of this equation, we find that the momentum of our blue ball before it collides with the pink ball is one kilogram meter per second.

Now, we can do the same thing for the pink ball. We can say that the momentum of the pink ball — we’ll call this 𝑝 subscript 𝑝 — before the collision is equal to its mass, which is one kilogram, multiplied by its velocity, which is zero meters per second. Because this ball is stationary. But then, any quantity multiplied by zero is simply zero. And hence, we can say that the momentum of the pink ball before the collision is zero. So then, what’s the total momentum of our system before the collision?

We’ll call this total momentum 𝑝 subscript tot. And because this’s before the collision — or in other words, this is the initial state of the system — we’ll add a little comma and put an 𝑖 there to represent the initial state. Now, the total momentum of the system is simply the momentum of all of the things within the system added up. And the things within the system are the blue ball and the pink ball. So, we add the momentum of the blue ball to the momentum of the pink ball. This ends up being one kilogram meter per second plus zero kilograms meters per second, which is altogether one kilogram meter per second.

So, at this point, we’ve seen what the total momentum of a system is. It is simply the sum of every object’s momentum. We can say very generally that the total momentum of a system consisting of lots of different objects is equal to the momentum of the first object plus the momentum of the second object plus the momentum of the third object plus so on and so forth. We add up the momentum of all the objects within that system.

Now, earlier, we made a pretty big deal of saying that the total momentum of the system that we calculated, one kilogram meter per second, was in fact the total momentum of the system initially, that is, before the two objects collided. Why did we do this? Well, it’s because this is where the conserved part of conservation of momentum is going to come into play. What does the word “conserved” mean? “Conserved” simply means that it remains constant over time. In other words then, the principle of conservation of momentum is telling us that the total momentum of an isolated system — we’ll come back to what “isolated” means in a minute — is the same at all times. It remains constant over time.

This means that for all times before the collision, the total momentum of the system stays the same; it does not change. But also, crucially, the total momentum of the system also stays the same after the collision. So, here is a diagram of our two balls colliding now. Let’s say that just after the collision, the blue ball now moves at a velocity 𝑣 one, and the pink ball now moves at a velocity 𝑣 two. Well, in this situation, what’s the total momentum of the system after the collision?

Well, the first thing we can do is to work out the momentum of the blue ball after the collision, which happens to be the mass of the blue ball, which is one kilogram still — that’s not changed — multiplied by its velocity, which we now know is 𝑣 one. And for the pink ball, we can say that the momentum after the collision is equal to one kilogram, which is still its mass, multiplied by its new velocity, which is 𝑣 two.

We don’t yet know the values of 𝑣 one and 𝑣 two, but the principle of conservation of momentum puts restrictions on these values. Specifically, if we once again say that the total momentum of our system, this time after the collision, so the final momentum, is equal to the momentum of the blue ball after the collision plus the momentum of the pink ball after the collision. And this is equal to one kilogram times 𝑣 one plus one kilogram times 𝑣 two. Then conservation of momentum is telling us that this total momentum after the collision has to be equal to the total momentum of the system before the collision.

In other words, one kilogram times 𝑣 one plus one kilogram times 𝑣 two has to all be equal to one kilogram meter per second. In other words, we can say that the total momentum of our system before the collision is equal to the total momentum of our system after the collision. And this is a pretty important statement to know because this is essentially the conservation of momentum in a nutshell. The total momentum of a system before some event, in this case that event being the collision, is equal to the total momentum after that event.

Now, one possible solution, in this case one possible set of velocities that the blue and pink ball could have after the collision, is that the blue ball stops moving entirely. So, 𝑣 one is equal to zero, which we may have seen happening if we’ve ever played pool. Because you can collide one ball against the other, and if you do it just the right way, then the first ball completely stops.

And in that case, we can see that the total momentum of the system before the collision, one kilogram meter per second, is equal to the momentum of the blue ball after the collision, which is one kilogram times zero meters per second. Because 𝑣 one is equal to zero as we said earlier. Plus the momentum of the pink ball, which is still going to be one kilogram times 𝑣 two. And this now allows us to calculate what 𝑣 two is. Because this term is zero, the total momentum of our system after the collision is simply one kilogram times 𝑣 two, in other words, the momentum of the pink ball.

In order to calculate what 𝑣 two is, we can divide both sides of the equation by one kilogram, thus actually making it cancel on both sides of the equation since we’ve got one kilogram in the numerator and denominator on both sides. And this leaves us with one meter per second is equal to 𝑣 two. In other words, after the collision, the blue ball is simply not moving. And 𝑣 two happens to be one meter per second, which, if we recall correctly, was the velocity of the blue ball before the collision.

And so, what’s actually happened here in this particular case is that our blue ball, which was initially coming in towards the pink ball with a velocity of one meter per second, has completely transferred all of its momentum now to the pink ball during the collision. Which means that after the collision, the blue ball completely stops; it has no momentum. And all of that momentum is instead given to the pink ball. But remember, the total momentum of the system is still conserved because the total momentum of the system is still one kilogram meter per second after the collision.

So, this means that objects within a system can transfer momentum to each other. It’s just that the overall system cannot gain or lose momentum; momentum is conserved. But there is one very important point to be made about this. This is only true if our system is isolated. There’s that word again. What does “isolated” mean? Well, isolated simply means that no external forces can be acting on our system. Remember how we said earlier that our system only consists of the blue ball and the pink ball and nothing else and how everything outside of this was external to the system?

Well, if anything outside the system exerts a force on any part of our system, then conservation of momentum no longer applies because it’s not an isolated system. For example, if at any point we were to come in and push one of the balls with our finger, then we would be exerting a force on it from outside the system because the finger is not part of the system. Hence, our system is no longer isolated. And conservation of momentum doesn’t apply here.

So, that’s a general overview of the principle of conservation of momentum. Let’s very quickly look at a general case for two objects colliding though. Because, in this case, we’ve looked at two objects colliding where both of them had the same mass. And we’ve looked at one particular scenario that does obey the conservation of momentum rule, where all of the momentum from the incoming object, in this case the blue ball, is transferred to the other object that it collides with. This isn’t always the case however. Because there are actually multiple different solutions to our conservation of momentum equation.

For example, if we had the same scenario once again where our blue ball has one kilogram of mass and our pink ball has also got one kilogram of mass. And the blue ball initially comes in with one meter per second, as it did earlier. And our pink ball was stationary at zero meters per second. And then, our blue ball collides with our pink ball. Then, another possible solution to the conservation of momentum equation is that both of the balls now move off toward the right, each with a velocity of 0.5 meters per second. In other words, in this particular case, the total momentum of the system is still conserved. Try calculating it for yourself.

But this time, instead of transferring all of its momentum to the pink ball when it collides with it, the blue ball actually only transfers half of its momentum to the pink ball and retains the other half. So, now, both of the balls are moving toward the right at a velocity of 0.5 meters per second. Therefore, we see that there are multiple different solutions to the conservation of momentum equation, assuming of course our system is isolated.

However, we need to be careful here. That doesn’t mean that the velocities of these two objects could be absolutely anything. They still must obey the conservation of momentum equation. It’s just that there are multiple different solutions to this equation.

So anyway, let’s generalize this a bit. Let’s say we’ve got two objects that are going to be colliding with each other. And the first object has a mass 𝑚 one, and the second object has a mass 𝑚 two, where the first object is moving with a velocity 𝑣 one and the second object is moving with a velocity 𝑣 two before the collision. Then the total momentum of the system initially, that’s before the collision, is equal to the mass of the first object multiplied by the velocity of the first object. This is the momentum of the first object. Plus the mass of the second object multiplied by the velocity of the second object. This is the momentum of the second object.

Now, let’s say they collide. And the first object moves now at a velocity of 𝑢 one, and the second object moves with a velocity of 𝑢 two. In other words, the objects’ masses are still the same as before, but their velocities have changed. In this case, the total momentum of the system after the collision, so the final total momentum of the system, is equal to the momentum of the blue object, which is 𝑚 one times 𝑢 one now, plus the momentum of the pink object, which is 𝑚 two times 𝑢 two. And so, the total momentum of an isolated system before must be equal to the total momentum of an isolated system after a collision according to the principle of conservation of momentum.

Mathematically then, we can say that the total momentum before is equal to the total momentum after. And if we instead substitute the right-hand sides of these equations for 𝑝 tot 𝑖 and 𝑝 tot 𝑓, then we get a really nice general equation for the conservation of momentum of a system consisting of two objects colliding. Assuming, of course, that it’s an isolated system. This equation is certainly not worth memorizing, but it is useful to know how to derive it. In other words, all the steps that we’ve gone through to calculate the general momentum of the system before the collision and the momentum of the system after then equating the two thus giving us this equation.

Now, there are a couple of important things that can be said here. Firstly, we can recall that momentum is a vector quantity. That means it has both magnitude or size and direction. And we definitely need to account for this directionality when considering conservation of momentum. So, let’s consider our two objects once again, the blue ball and the pink ball. The blue ball has a mass 𝑚 one, and the pink ball has a mass 𝑚 two. But this time, the blue ball is moving toward the right and the pink ball is moving toward the left. They’re moving closer to each other.

Let’s say that the velocity of the blue ball is 𝑣 one and the velocity of the pink ball is 𝑣 two. Or more specifically, the velocity of the blue ball is 𝑣 one toward the right, and the velocity of the pink ball is 𝑣 two toward the left. At this point, then, we need to decide a convention. We can choose to say that anything moving toward the right is moving in the positive direction, in which case anything moving toward the left must be moving in the negative direction. Or vice versa, we could choose the left to be positive and right to be negative, but we do have to choose at this point.

So, let’s stick with the convention we’ve drawn here. And let’s find out the total momentum of the system here. Of course, it’s before the collision, but that’s not really relevant to us right now. And we can say that this total momentum of the system is equal to the momentum of the blue ball, which is equal to 𝑚 one, the mass of the blue ball, multiplied by 𝑣 one, the velocity of the blue ball. Plus the momentum of the pink ball, which is 𝑚 two, the mass of the pink ball, multiplied by negative 𝑣 two, which is the velocity of the pink ball. And this is because the velocity of the pink ball has a magnitude which we’ve called 𝑣 two, and it’s moving toward the left. And anything moving toward the left is moving in the negative direction.

So, we’re considering the directionality of the velocities of our objects because velocity is also a vector quantity. And when we do take into consideration the directions of the velocities, that automatically accounts for the direction of the momentum. But it is something that we need to be conscious of.

Let’s look at a special case of this. Let’s say that both the blue ball and the pink ball have a mass of one kilogram each and are both moving at one meter per second. Except, the blue ball is moving toward the right, and the pink ball is moving toward the left. So, the momentum of the blue ball before the collision is equal to one kilogram times one meter per second, which is equal to one kilogram meter per second. And the momentum of the pink ball before the collision is equal to one kilogram times negative one meter per second. Because, remember, we decided that anything moving toward the left is moving in the negative direction. And this ends up being negative one kilogram meter per second.

So, what’s the total momentum of our system before the collision? Well, it happens to be the momentum of the blue ball plus the momentum of the pink ball, which is one kilogram meter per second plus negative one kilogram meter per second. Those add together to give zero kilogram meters per second. In other words, the total momentum of our system before the collision is actually zero, which makes sense. The blue ball has a certain amount of momentum toward the right, and the pink ball has exactly the same amount of momentum toward the left. So, they cancel each other out.

So, now, let’s imagine that the two balls collide. Conservation of momentum tells us that the total momentum of the system after the collision must still be zero kilogram meters per second. But there are multiple different ways to achieve this. One possible solution is that, upon collision, both the balls remain stationary; they just stop moving. In that case, each ball has a velocity of zero meters per second. And this gives us a total momentum of our system of zero kilogram meters per second, just as we expect. Try calculating this total momentum for yourself for this particular scenario.

Another possible solution is that these two balls move off in opposite directions to the directions in which they were earlier moving. And they both move off at the same speed. So, for example, the blue ball may now start moving toward the left at 0.5 meters per second, and the pink ball may start moving toward the right at 0.5 meters per second. This still means that the new momentum of the blue ball after the collision is equal in magnitude to the new momentum of the pink ball after the collision. Even though before the collision both balls were moving at one meter per second in opposite directions and after the collision they’re actually both moving at a slower speed.

Now, this may happen because when the balls collide, some of the energy that they had before the collision gets converted to, say, sound energy when we hear a click when the balls collide or heat energy because the balls heat up when they collide. But what really matters is that the momentum of the total system before the collision is conserved. And so, the total momentum of the system afterwards is zero as well because the momenta of the two balls individually after they’ve collided cancel each other out in this case.

So, at this point, we’ve looked at various different scenarios. And we’ve also seen how the principle of conservation of momentum can be applied to different collisions to understand them better. So, having said all of this, let’s summarize what we’ve talked about in this lesson.

We firstly recall that the momentum of an object 𝑝 is given by 𝑝 is equal to 𝑚𝑣, where 𝑚 is the mass of the object and 𝑣 is its velocity. We also saw that momentum is a vector quantity. We learned that the principle of conservation of momentum states that the total momentum of an isolated system is constant at all times. An isolated system is one on which no external forces act. And finally, conservation of momentum can be applied to collisions between objects to understand how the objects’ velocities change as a result of the collision.