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.
