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Video: Determining the Change of Velocity of an Object Required to Obtain a Particular Momentum

Parth Gharfalkar

A cricket ball of mass 160 g has a constant velocity of 10 m/s. A golf ball of mass 40 g has the same velocity as the cricket ball. What velocity change must the cricket ball undergo to have the same momentum as the golf ball?

05:43

Video Transcript

A cricket ball of mass one 160 grams has a constant velocity of 10 meters per second. A golf ball of mass 40 grams has the same velocity as the cricket ball. What velocity change must a cricket ball undergo to have the same momentum as the golf ball?

Okay, so let’s begin by underlining the important information given to us in the question. Now we know we’ve got two balls. The first one is a cricket ball of mass 160 grams. And it has a constant velocity of 10 meters per second. The second is a golf ball with a mass of 40 grams. And it has the same velocity. What we need to do is find out the velocity change that the cricket ball must undergo to have the same momentum as the golf ball.

So if we’re to draw a diagram, in this case really badly, here’s our cricket ball with the mass 𝑚 sub c of 160 grams and it’s travelling at a velocity 𝑉 sub c at 10 meters per second. We can also draw in our golf ball which has a mass 𝑚 sub g of 40 grams and a velocity 𝑉 sub g of 10 meters per second, same as the cricket ball. Now this question is quite intricate. What it wants us to do is to find the change in velocity that the cricket ball must go through in order to have the same momentum as the golf ball. So first of all, why don’t we find out what the momentum of the cricket ball and the momentum of the golf ball actually is.

Using the definition that momentum is equal to the mass multiplied by the velocity, we can say that the momentum of the cricket ball now, capital 𝑀 sub c, is equal to lowercase 𝑚 sub c, which is the mass, multiplied by 𝑉 sub c, which is the velocity. And we can substitute in the values of 160 grams and 10 meters per second. This ends up giving us a momentum for the cricket ball of 1600 gram meters per second. By the way, you may have noticed that we’ve kept using grams rather than kilograms which is the standard unit of mass. But in this case, it doesn’t matter. As long as we also use grams for the golf ball, we can find the momentum in gram meters per second and compare the two rather than converting the mass to kilograms.

So now we know the initial momentum of the cricket ball. Which means we can find the momentum of the golf ball. Once again, the momentum — this time of the golf ball, capital 𝑀 sub g — is equal to the mass of the golf ball, lower case 𝑚 sub g, multiplied by the velocity, 𝑉 sub g. Going through the motions once again, we see that the momentum of the golf ball, capital 𝑀 sub g, happens to be 400 gram meters per second.

Now the question wants us to find the velocity change that the cricket ball must go through to have the same momentum as the golf ball. So let’s ignore the velocity change bit for now. And let’s just discuss the bit which talks about the same momentum. We’ve seen that the momentum of the golf ball is 400 gram meters per second. And the question wants the cricket ball to have the same momentum of 400 gram meters per second. That is, the question wants the new cricket ball momentum, capital 𝑀 sub c comma new, to be 400 gram meters per second. So the momentum of the cricket ball started out at 1600 gram meters per second. And it has to be reduced to 400 gram meters per second.

Now there are two ways of doing this. In order to reduce momentum, you could either lower the value of the mass, that is basically chop the cricket ball into pieces, or you could make the cricket ball lose some speed. And it’s the second scenario that’s important in this question because, obviously, the question doesn’t talk about slicing up the cricket ball. But what it does talk about is changing velocity. Which means that we can say that the new momentum, capital 𝑀 sub c comma new, is equal to 𝑚 sub c, which is the same mass of the cricket ball cause the mass doesn’t change, multiplied by a new velocity, 𝑉 sub c comma new. And we could find this new velocity, 𝑉 sub c comma new, by dividing both sides of the equation by the mass of the cricket ball, lowercase 𝑚 sub c. And that happens to be 400 gram meters per second, the new momentum of the cricket ball, divided by 160 grams, which is still the mass of the cricket ball.

Evaluating that fraction tells us that the new velocity, 𝑉 sub c comma new, is 2.5 meters per second. But is that our final answer? No, it is not. We need to read the question carefully one more time. What we’ve been told to find is the velocity change that the cricket ball must go through in order to have this new momentum. So we need to work out this velocity change. We don’t just want the final velocity. We want the velocity change. What is a velocity change? We need to work out a quantity Δ𝑉. Now Δ is used to represent a change. And we want the velocity change. And this velocity change is given by the difference between the final velocity of the cricket ball and the initial velocity of the cricket ball. In other words, it’s given by 𝑉 sub c comma new minus 𝑉 sub c. Because initially, it was travelling at 𝑉 sub c. That’s 10 meters per second. And now it’s travelling at 𝑉 sub c comma new. That’s two and a half meters per second. So we need to work out what the difference between these two is, to work out what the change in velocity is.

So plugging in these values, we get two and a half meters per second minus 10 meters per second. And so, our final answer is that the change in velocity of the cricket ball is minus 7.5 meters per second. Or, in other words, the cricket ball loses 7.5 meters per second of speed so that it can have the same momentum as the golf ball. This makes some physical sense. Since the cricket ball is much heavier than the golf ball, in order to have the same momentum, it must be travelling a lot slower. It cannot be travelling at the same speed as the golf ball.