Question Video: Comparing the Momenta of Objects Physics

Video Transcript

How many times greater is the momentum of a 1500-kilogram-mass hippo moving at 0.25 meters per second than a 75-gram-mass bird flying at 10 meters per second?

Okay, so in this question, we need to work out how many times greater the momentum of a 1500-kilogram hippo is when that hippo is moving at 0.25 meters per second compared to the momentum of a 75-gram-mass bird flying at 10 meters per second. The first thing that we can do is to work out the momentum of the hippo and the momentum of the bird. So how do we do that?

Well, we can recall that the momentum of an object, 𝑃, is found by multiplying the mass of the object, 𝑚, by the velocity of the object, 𝑣. Now we know the masses and the velocities of both the hippo and the bird. So we can work out the momentum of the hippo and the momentum of the bird.

Let’s start by working out the momentum of the hippo. Let’s call this 𝑃 sub ℎ for momentum of the hippo. Well, 𝑃 sub ℎ is going to be 𝑚 sub ℎ, the mass of the hippo, multiplied by 𝑣 sub ℎ, the velocity of the hippo. And we know that the mass of the hippo is 1500 kilograms and its velocity is 0.25 meters per second. So when we multiply 1500 by 0.25, the numerical value ends up being 375.

But what are the units of momentum? Well, we found momentum by multiplying mass by the velocity. Now the mass was in kilograms, kg, and the velocity was in meters per second. So the units of momentum are going to be kilograms meters per second.

And at this point, we know the momentum of the hippo. So we can work out the momentum of the bird. Let’s call this 𝑃 sub 𝑏 for the momentum of the bird. And as we expect, this is 𝑚 sub 𝑏, the mass of the bird, multiplied by 𝑣 sub 𝑏, the velocity of the bird. But now we have a problem, because the mass of the bird has been given to us in grams, whereas the mass of the hippo had been given to us in kilograms.

In order for us to stay consistent, we need to compare the momentum of the bird with the momentum of the hippo using the same units. In other words, we want to find the momentum of the bird in kilograms meters per second, not grams meters per second. And the way to do this is to convert the mass of the bird, 75 grams, into kilograms. So how do we do that?

Well, we start by recalling that one kilogram is equal to 1000 grams. That’s what the prefix “kilo” means. “Kilo” means 1000, which means that what we can do is to divide this equation by 1000. And that’s exactly what we’ve done here. We divided both sides by 1000.

What that leaves us with is that 1000 of a kilogram is equal to one gram. But the mass of the bird is 75 grams. So we can multiply this equation by 75. And of course, we do this to both sides of the equation. And what that gives us is that 75 divided by 1000 kilograms is equal to 75 grams. But we can evaluate the fraction on the left-hand side to give us 0.075 kilograms is equal to 75 grams. Therefore, that’s the mass of the bird in kilograms. It’s 0.075 kilograms. And so that’s the numerical value we’ll use.

Hence, we find that 𝑃 sub 𝑏, the momentum of the bird, is equal to the mass of the bird in kilograms — that’s 0.075 — multiplied by the velocity of the bird — that’s 10 meters per second. So when we multiply 0.075 by 10, we find that the momentum of the bird is 0.75 kilograms meters per second as we expected.

So now we’ve found the momentum of the hippo and the momentum of the bird. But what we need to do in this question is to find out how many times greater the momentum of the hippo is compared to the momentum of the bird. One way to do this, which is fairly intuitive, is to say that the momentum of the hippo, 𝑃 sub ℎ, is 𝑛 times larger than the momentum of the bird, 𝑃 sub 𝑏. Or even more simply, the momentum of the hippo, 𝑃 sub ℎ, is 𝑛 times the momentum of the bird.

In this equation, it’s not particularly clear whether the momentum of the hippo is larger than the momentum of the bird. But to us, that doesn’t matter. We’ve already been told in the question that the momentum of the hippo is larger than the momentum of the bird. So all we need to do is to find out the value of 𝑛, because 𝑃 sub ℎ is 𝑛 times 𝑃 sub 𝑏.

So to find 𝑛, what we can do is divide both sides of the equation by 𝑃 sub 𝑏. What that results in is 𝑃 sub 𝑏 canceling on the right-hand side, leaving us with 𝑛 is equal to 𝑃 sub ℎ over 𝑃 sub 𝑏, which is this right-hand side here. And since we already know what 𝑃 sub ℎ and 𝑃 sub 𝑏 are, all what we need to do is to substitute them in.

And remember, we can do that because we’ve got both 𝑃 sub ℎ and 𝑃 sub 𝑏 in the same set of units, kilograms meters per second. This wouldn’t have been possible if we’d found 𝑃 sub 𝑏 in grams meters per second. So it’s a good thing we converted the mass of the bird into kilograms.

Anyway, let’s sub the values in. As we said, 𝑛 is equal to 𝑃 sub ℎ, which is 375 divided by 𝑃 sub 𝑏, which is 0.75. And when we evaluate this fraction, we find that 𝑛 is exactly equal to 500. In other words, the momentum of the hippo is 500 times larger. And so we have our final answer. The momentum of the hippo is 500 times larger than the momentum of the bird.