Video: Conservation of Momentum

During a game of pool, a cue ball that has a mass of 170 g moves along a pool table at a constant speed of 80 cm/s. The cue ball hits another ball which has a mass of 160 g and is not moving as shown in the diagram. After the collision, the cue ball does not move. What constant speed, in centimeters per second, does the 160 g mass ball move at after the collision?

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Video Transcript

During a game of pool, a cue ball that has a mass of 170 grams moves along a pool table at a constant speed of 80 centimeters per second. The cue ball hits another ball which has a mass of 160 grams and is not moving as shown in the diagram. After the collision, the cue ball does not move. What constant speed, in centimeters per second, does the 160-gram mass ball move at after the collision?

Okay, so this question, we’ve got a cue ball which has a mass of 170 grams and another ball — let’s call this the orange ball — which has a mass of 160 grams. We’ve been told that, before the collision between the two balls, the cue ball is moving at 80 centimeters per second towards the orange ball. And then after the collision, the cue ball does not move.

So before we do anything else, let’s draw another diagram to show what happens after the collision. Well, after the collision, the masses of the balls don’t change because we haven’t been given any indication of that happening. But also the 170-gram cue ball does not move. This means that the orange ball must start moving towards the right. This is because of the law of conservation of momentum.

This law tells us that the total momentum in an isolated system is conserved. In other words, if we add up all the momentum of all of the components of a system — so that’s the total momentum — then that total momentum stays the same at all times. But this is only true if the system is isolated. In other words, there are no external forces acting on any components of the system.

Well, in this situation, we’ve got a cue ball that’s already moving at 80 centimeters per second towards the orange ball. So from the time that we start considering the system, there are no external forces acting on the system. Of course, there would’ve initially been a force on the cue ball to get it moving at 80 centimeters per second in the first place. But from the time that we start considering the system, the cue ball is already moving at 80 centimeters per second. So our system that consists of the cue ball, the orange ball, and the pool table has no external forces or any other external influences acting on it. And hence, we do have an isolated system here. This means that the total momentum of the system must be conserved. It must be constant at all times. And this is true for before the collision and after the collision.

Now let’s quickly recall that momentum is defined as the mass of an object multiplied by the velocity with which that object is moving. And based on this definition, we can see that, initially, the cue ball is the only thing with any momentum in this entire system. This is because only the cue ball has velocity, whereas the orange ball is not moving. So it does not have any momentum.

However, after the collision, we’ve been told that the cue ball does not move. So all of that momentum that the cue ball had before the collision must therefore have been transferred to the orange ball. And hence, the orange ball must start moving in the same direction, because remember momentum and velocity are vector quantities. In other words, they have both magnitude and direction. And because the cue ball was initially moving towards the right, the momentum of the cue ball was towards the right as well. This means that, in order for momentum to be conserved after the collision, the orange ball must also move towards the right.

So now that we’ve understood a little bit about what happens in the system physically, let’s try and calculate what the velocity of this orange ball should be. Let’s firstly call this velocity 𝑣 subscript 𝑜 for the velocity of the orange ball. Then we can start by calculating the total momentum of the system before the collision. Let’s call this total momentum 𝑃 subscript bef for the momentum before. And we know that it’s equal to the momentum of the cue ball plus nothing else because that’s the only momentum in the system at that point.

Therefore, it’s going to be equal to the mass of the cue ball, which is 170 grams, multiplied by the velocity with which it’s moving, which is 80 centimeters per second. So that’s the total momentum of the entire system before the collision.

Now let’s find an expression for the total momentum of the entire system after the collision, 𝑃 subscript aft. Well, this is going to be equal to this time the mass of the orange ball, because now that’s the one moving. And hence, it’s 160 grams. And we multiply this by the velocity 𝑣 subscript 𝑜 because we don’t know what 𝑣 𝑜 is yet.

Now the reason that we didn’t need to use the mass of the cue ball in this situation is because the momentum of the cue ball is simply 170 grams multiplied by zero because it’s not moving. And so the momentum of the cue ball is zero. Hence, the momentum of the entire system, 𝑃 subscript aft, is simply the momentum of the orange ball.

Now the law of conservation of momentum tells us that the two momenta before and after the collision must be equal because this is how the total momentum is conserved. And hence, we can equate the two right-hand sides of our equations. Hence, we get that 170 grams multiplied by 80 centimeters per second is equal to 160 grams multiplied by 𝑣 𝑜. Then we can divide both sides of the equation by 160 grams. So it cancels on the right. And on the left, we can see that the unit of grams in the numerator cancels with the unit of grams in the denominator. What this leaves us with is 𝑣 𝑜 on the right-hand side and some number with the unit centimeters per second on the left-hand side.

Now this is perfect because we’ve been asked to find the constant speed in centimeters per second that the 160-gram mass ball — that’s the orange ball — moves at. And what we’re finding here is the speed with which the ball moves. We might think that we found the velocity. But we actually know that the velocity consists of the speed, which is what we’re calculating here, and the direction, which we know is to the right. So the calculation is simply going to give us the magnitude of the velocity, 𝑣 𝑜, or in other words the speed with which the orange ball is moving.

Therefore, evaluating everything on the left-hand side of the equation, we find that this is equal to 85 centimeters per second. And so we have our final answer. The 160-gram mass ball after the collision moves at a constant speed of 85 centimeters per second.

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