Video: Momentum Change of a Body due to a Collision

A tennis ball with a mass of 60 g is thrown at a wall, hitting it while moving at a speed of 15 m/s. The ball bounces back from the wall with a speed 10 m/s. What is the magnitude of the tennis ball’s net momentum change due to the collision?

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Video Transcript

A tennis ball with a mass of 60 grams is thrown at a wall, hitting it while moving at a speed of 15 metres per second. The ball bounces back from the wall with a speed 10 metres per second. What is the magnitude of the tennis ball’s net momentum change due to the collision?

Alright, so, in this question, we’re considering a tennis ball. So, let’s say that this is our tennis ball. And we’ve been told that it collides with a wall. So, here’s the wall that the tennis ball collides with. Now we’ve also been told that while the tennis ball hit the wall, it was moving with a speed of 15 metres per second. In other words, before the collision, the tennis ball must have been moving at 15 metres per second towards the wall.

And then, we’ve been told that the ball bounces back from the wall with a speed 10 metres per second. In other words, now it’s travelling this way, away from the wall, at 10 metres per second. So, given all of this information, we need to find the magnitude of the ball’s net momentum change due to this collision with the wall.

So, to do this let’s start by recalling what we mean by momentum. The momentum 𝑝 of an object is defined as the mass of the object multiplied by the velocity with which it’s travelling. And so, at different points along the ball trajectory, we could work out its momentum. For example, we could work out the momentum of the tennis ball before it collides with the wall. And we could work out the momentum after it collides with the wall.

However, we don’t need to find the ball’s momentum. We need to find the ball’s net momentum change. Now the word net basically means overall. And so, we need to find the overall change in momentum of the ball due to the collision with the wall. So, to find this net, or overall, momentum change, we can recall that the change in any quantity is given by finding the final value of that quantity minus its initial value.

So, in our case, to find the net momentum change due to the collision, we can say that this change in momentum, which we’ll call 𝛥𝑝, is equal to the final momentum of the tennis ball, which we’ll call 𝑝 subscript final, minus the initial momentum, which we’ll call 𝑝 subscript initial. Now from this, we can say that the change in momentum is equal to the mass multiplied by the final velocity, 𝑉 subscript final we’ll call it, of the tennis ball minus the mass of the tennis ball multiplied by the initial velocity.

Now it’s important to know that throughout this entire sequence of events, the mass of the ball is not changing. And hence, we’ve used the same value 𝑚 for the mass of the tennis ball. If the mass of the ball had changed as well, then we would have to put subscripts here and here as final and initial as well. However, luckily, we don’t need to worry about that.

Also, in the question, we can see that the mass of the tennis ball 𝑚 has been given to us as 60 grams. However, we haven’t been given the final and initial velocities of the tennis ball. We’ve simply been given the final and initial speeds of the tennis ball. Now the difference between speed and velocity is that velocity is a vector quantity. And therefore, as well as addressing the magnitude of the velocity, or the size of the velocity, we also need to address the direction in which the object is travelling.

And luckily, we can do this based on the way the question is worded. We’ve been told that the initial speed of the ball is in the direction such that the ball is travelling towards the wall. Because, otherwise, how would it hit the wall? And similarly, the final speed of the ball, 10 metres per second, is in the direction away from the wall. Because we’ve been told that the ball bounces back from the wall.

Now it’s just coincidence that we’ve drawn the wall to the right of the tennis ball. We could have drawn it in any orientation we wanted. However, as we’ll see soon, that doesn’t really matter. All that matters is that we know the ball is initially travelling at 15 metres per second towards wherever the wall is. And then, the final velocity is 10 metres per second away from the wall.

Now at this point, we can also arbitrarily choose to say that the ball moving away from the wall is moving in a positive direction. And therefore, if the ball is moving in the opposite direction, that is towards the wall, then it’s moving in the negative direction.

As well as this, before we go on to calculate the change in momentum of the tennis ball, we need to remember to convert this 60-gram mass of the tennis ball into base units. We can recall that the base unit of mass is kilograms. Whereas the quantity we’ve been given in the question is in grams. So, to combat this, we can recall that one gram is equivalent to one thousandth of a kilogram. And therefore, 60 grams is equivalent to 60 thousandths of a kilogram, or in other words, 0.06 kilograms.

So, now we can go on to calculate the change in momentum of the tennis ball. This change in momentum is equal to the mass of the tennis ball, which is 0.06 kilograms, multiplied by the final velocity of the ball, which we said is 10 metres per second. And because it’s travelling towards the left, that’s positive. And so, we say the final velocity is 10 metres per second. And from this, we need to subtract the mass of the tennis ball, once again 0.06 kilograms, multiplied by the initial velocity of the tennis ball, which is 15 metres per second, but in the negative direction. And so, we say the initial velocity is negative 15 metres per second.

And then. when we evaluate this whole right-hand side of the equation, we find that the change in momentum of the tennis ball due to the collision is 1.5 kilograms metres per second. And because this whole quantity is positive, we can see that the change in momentum is in this direction. That’s the positive direction.

And that makes sense. Because in order for the ball to go from travelling at 15 metres per second to the right to travelling at 10 metres per second to the left, it had to not only decelerate until it was not moving at all, but it also, then, had to accelerate to the left. And hence, its velocity has not only gone from 15 metres per second this way to zero, it’s actually gone to 10 metres per second in the opposite direction. And so, the momentum of the ball has changed from being in this direction all the way to being in this direction.

Now at this point. it’s also worth noting what would happen if we had switched the sign conventions, if we’d said that towards the right was positive and if towards the left was negative. Well, in that situation, we could complete the calculation again. And we’d see that the change in momentum was negative. In other words, the change in momentum of the tennis ball would be negative 1.5 kilograms metres per second in this direction. Or in other words, it would be 1.5 kilograms metres per second in this direction, same as before. The momentum is increasing a lot in this direction. And that’s exactly what we get, regardless of which sign convention we use.

But then, what do we give as a final answer? Do we give positive 1.5 kilograms metres per second or negative 1.5 kilograms metres per second? It almost seems like the answer should depend on what sign convention we choose. Well, no, it shouldn’t because we’ve been asked to find the magnitude of the tennis ball’s net momentum change. And the magnitude simply means size. And so, we don’t need to worry about signs at all. We simply give the positive value.

Because let’s think about it this way. The negative sign or the positive sign in front of the 1.5 kilograms metres per second will simply tell us the direction in which the momentum is changing. And momentum, or change in momentum, is a vector quantity. And so, if we’re only asked to give its magnitude, then we only need to worry about the size of the change in momentum. Therefore, we can say that the magnitude of the tennis ball’s net momentum change due to the collision is 1.5 kilograms metres per second.

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