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.
