Video Transcript
A bowling ball with a mass of 12.5 kilograms and a momentum of 25 kilograms-meters per second collides for 0.25 seconds with a stationary bowling pin of mass 7.5 kilograms. After the collision, the pin moves at 2.5 meters per second in the same direction that the ball moved before the collision. The ball slows to a speed of 0.5 meters per second. What is the change of the momentum of the pin? What is the change of the momentum of the ball? What is the force applied to the pin? What is the force applied to the ball?
Okay, so this is a long question with lots and lots of information in it. We need to underline all of the important bits, including all the numerical values we’ve been given so we don’t miss anything out. So we’ve got a bowling ball, and it’s got a mass of 12.5 kilograms. It also has a momentum of 25 kilograms-meters per second before its collision.
The collision itself last for 0.25 seconds, and that happens with a stationary bowling pin. The bowling pin has a mass of 7.5 kilograms. After the collision, the pin moves at 2.5 meters per second. And it moves in the same direction that the ball moved before the collision. Finally, the ball slows to a speed of 0.5 meters per second.
Now this amount of information can seem really, really overwhelming. So the first thing to do in a situation like this is to draw a diagram. There! That doodling must have sure calmed our nerves about this question. No I’m kidding! Drawing a diagram is genuinely useful for physics, and we’ll see how in a second.
So we’ve got this bowling ball. And before it collides with the pin, it’s moving with a momentum of 25 kilograms-meters per second. We also know that it has a mass of 12.5 kilograms. We know the pin has a mass of 7.5 kilograms. And since it’s stationary, we won’t draw any arrows on it to show that it’s not moving anywhere. All of this is before the collision. We can also draw a diagram to show what’s happening after the collision.
So here are the bowling ball and the pin. This time the pin’s being knocked forward. And it’s moving in the direction that the ball was initially moving in. It’s moving at 2.5 meters per second. That’s its velocity. We know the ball is also still moving, but this time it’s slowed to 0.5 meters per second. And all of this is after the collision. So let’s look at the very first question. What is the change of the momentum of the pin?
We can find the pin’s momentum by recalling that the momentum, 𝑝, is given by multiplying the mass, 𝑚, by the velocity, 𝑣. So in the case of the bowling pin, we can work out the initial momentum and the final momentum; that is, the momentum before the collision and the momentum after the collision. So let’s start with the momentum before the collision. We’ll call that 𝑝 sub 𝑏 for momentum sub before. And since it’s equal to the mass multiplied by the velocity, then we have 7.5, which is the mass multiplied, by zero, which is its initial velocity.
But anything multiplied by zero is zero. So the pin’s initial momentum is zero kilograms-meters per second. Let’s now work out the final momentum of the pin; that is, 𝑝 sub 𝑎 for momentum after. This time, the mass is still 7.5 kilograms, but the velocity is 2.5 kilograms. Multiplying these together, we get 18.75 kilograms-meters per second. But what we’re trying to work out is the change of the momentum of the pin. We can label that as Δ𝑝 because Δ means change in.
And the change in momentum is simply given by the momentum after the collision minus the momentum before the collision. And this is simply 18.75 minus zero, which turns out to be 18.75 kilograms-meters per second. And since the change in momentum is in the forward direction, that’s the direction that the ball was travelling in earlier, the sign on the change in momentum is positive. So our final answer is that the change of the momentum of the pin is 18.75 kilograms-meters per second.
Moving on to the next bit, what is the change of the momentum of the ball? Well we can apply conservation of momentum here. Conservation of momentum states that the total momentum of a system after a collision is the same as the total momentum of the system before the collision, provided no external forces act on the system. Now in this case, we don’t have any external forces because there’s nothing from the outside world acting on the bowling pin and bowling ball combo. Instead, they’re merely interacting with each other. So conservation of momentum must hold.
Now we’ve already seen that the bowling pin has gained 18.75 kilograms-meters per second in momentum. Since the bowling ball is the only other thing in the system, that means that in order for conservation of momentum to be a thing, the bowling ball must have lost 18.75 kilograms-meters per second. Therefore, the change in the momentum of the ball is negative 18.75 kilograms-meters per second. We can also very quickly work this out using the same method that we used earlier.
The momentum of the bowling ball before the collision is 25 kilograms-meters per second. We’re given that in the question. The momentum after the collision happens to be 12.5 kilograms, which is the mass of the bowling ball, multiplied by 0.5 meters per second, which is its speed after the collision. And this momentum happens to be 6.25 kilograms-meters per second. So the change of the momentum of the ball, Δ𝑝, is once again given by 𝑝 sub 𝑎 minus 𝑝 sub 𝑏, the momentum after the collision minus the momentum before, which happens to be 6.25 kilograms-meters per second minus 25 kilograms-meters per second.
And as we expect, we get negative 18.75 kilograms-meters per second. It’s lost momentum so the answer is negative. And in other words, the ball has slowed down. Hence, its momentum is lower after the collision than it was before. So happily, we can move on to the next part of the question. This bit asks us what is the force applied to the pin. Now in this question, we’re working with momenta, but we’ve also been given one other piece of information that we’ve so far ignored.
We’ve been told that the collision between the ball and the pin lasted for 0.25 seconds. This is useful because now we’ve got momentum and the time for which the collision occurred. This means that we can look towards impulses to help us with our calculation. An impulse occurs when a force is being applied to an object for a very short period of time. So an impulse is defined as the force, 𝐹, multiplied by the short period of time, Δ𝑡. But this impulse is also equal to the change in momentum of the object, Δ𝑝. In this case, we’re trying to work out what the force is.
We know that the short period of time for which the force is applied is 0.25 seconds. And we’ve already calculated that the change in momentum is 18.75 kilograms- meters per second. By the way, the force being applied on the bowling pin is of course from the ball colliding with it. Anyway, so all we need to do is to rearrange the impulse equation, which looks something like this. By dividing both sides of the equation by Δ𝑡, we’ve managed to isolate 𝐹 on the left-hand side. We can then substitute in our values of 18.75 for Δ𝑝 and 0.25 for Δ𝑡. This leaves us with the force of 75 newtons. And so our final answer to that question is that the force applied to the pin is 75 newtons.
Now, the final question is asking us what is the force applied to the ball. This answer we can find in multiple ways. One of the ways is to use Newton’s third law of motion. Newton’s third law states that when a force is exerted on an object, it exerts an equal and opposite force on the object exerting the force. So in our case, when the bowling ball exerts a force on the bowling pin, the bowling pin exerts an equal force in the opposite direction on the bowling ball.
So if the bowling ball exerts a 75-newton force to the right on the bowling pin, then the bowling pin exerts a 75-newton force to the left on the bowling ball. This is the essence of Newton’s third law. Therefore, the force on the bowling ball is 75 newtons, but to the left. And so the answer is that the force applied to the ball is negative 75 newtons. This force is what is causing the ball to slow down. Initially, it was travelling at a certain speed to the right. And then it slowed down to 0.5 meters per second.
We’ve been told that the ball slows down in the question, and this makes sense. Also, we could’ve found this negative 75-newton Force by the same method that we used to find the 75-newton force on the pin, by using impulses. However, we can avoid all the working out by using Newton’s third law. And this question serves to demonstrate that quite often there are multiple ways of working out a solution. Therefore, if we can find the right method, we can save a lot of time. So to recap, the change in momentum of the pin is 18.75 kilograms-meters per second. The change in momentum of the ball is negative 18.75 kilograms-meters per second. The force applied to the pin is 75 newtons. And the force applied to the ball is negative 75 newtons.
