Video Transcript
During a game of pool, a cue
ball that has a mass of 170 grams moves along a pool table at a constant speed
of 80 centimeters per second. The cue ball hits another ball
which has a mass of 160 grams and is not moving as shown in the diagram. After the collision, the cue
ball does not move. What constant speed, in
centimeters per second, does the 160-gram mass ball move at after the
collision?
Okay, so this question, we’ve
got a cue ball which has a mass of 170 grams and another ball — let’s call this
the orange ball — which has a mass of 160 grams. We’ve been told that, before
the collision between the two balls, the cue ball is moving at 80 centimeters
per second towards the orange ball. And then after the collision,
the cue ball does not move.
So before we do anything else,
let’s draw another diagram to show what happens after the collision. Well, after the collision, the
masses of the balls don’t change because we haven’t been given any indication of
that happening. But also the 170-gram cue ball
does not move. This means that the orange ball
must start moving towards the right. This is because of the law of
conservation of momentum.
This law tells us that the
total momentum in an isolated system is conserved. In other words, if we add up
all the momentum of all of the components of a system — so that’s the total
momentum — then that total momentum stays the same at all times. But this is only true if the
system is isolated. In other words, there are no
external forces acting on any components of the system.
Well, in this situation, we’ve
got a cue ball that’s already moving at 80 centimeters per second towards the
orange ball. So from the time that we start
considering the system, there are no external forces acting on the system. Of course, there would’ve
initially been a force on the cue ball to get it moving at 80 centimeters per
second in the first place. But from the time that we start
considering the system, the cue ball is already moving at 80 centimeters per
second. So our system that consists of
the cue ball, the orange ball, and the pool table has no external forces or any
other external influences acting on it. And hence, we do have an
isolated system here. This means that the total
momentum of the system must be conserved. It must be constant at all
times. And this is true for before the
collision and after the collision.
Now let’s quickly recall that
momentum is defined as the mass of an object multiplied by the velocity with
which that object is moving. And based on this definition,
we can see that, initially, the cue ball is the only thing with any momentum in
this entire system. This is because only the cue
ball has velocity, whereas the orange ball is not moving. So it does not have any
momentum.
However, after the collision,
we’ve been told that the cue ball does not move. So all of that momentum that
the cue ball had before the collision must therefore have been transferred to
the orange ball. And hence, the orange ball must
start moving in the same direction, because remember momentum and velocity are
vector quantities. In other words, they have both
magnitude and direction. And because the cue ball was
initially moving towards the right, the momentum of the cue ball was towards the
right as well. This means that, in order for
momentum to be conserved after the collision, the orange ball must also move
towards the right.
So now that we’ve understood a
little bit about what happens in the system physically, let’s try and calculate
what the velocity of this orange ball should be. Let’s firstly call this
velocity 𝑣 subscript 𝑜 for the velocity of the orange ball. Then we can start by
calculating the total momentum of the system before the collision. Let’s call this total momentum
𝑃 subscript bef for the momentum before. And we know that it’s equal to
the momentum of the cue ball plus nothing else because that’s the only momentum
in the system at that point.
Therefore, it’s going to be
equal to the mass of the cue ball, which is 170 grams, multiplied by the
velocity with which it’s moving, which is 80 centimeters per second. So that’s the total momentum of
the entire system before the collision.
Now let’s find an expression
for the total momentum of the entire system after the collision, 𝑃 subscript
aft. Well, this is going to be equal
to this time the mass of the orange ball, because now that’s the one moving. And hence, it’s 160 grams. And we multiply this by the
velocity 𝑣 subscript 𝑜 because we don’t know what 𝑣 𝑜 is yet.
Now the reason that we didn’t
need to use the mass of the cue ball in this situation is because the momentum
of the cue ball is simply 170 grams multiplied by zero because it’s not
moving. And so the momentum of the cue
ball is zero. Hence, the momentum of the
entire system, 𝑃 subscript aft, is simply the momentum of the orange ball.
Now the law of conservation of
momentum tells us that the two momenta before and after the collision must be
equal because this is how the total momentum is conserved. And hence, we can equate the
two right-hand sides of our equations. Hence, we get that 170 grams
multiplied by 80 centimeters per second is equal to 160 grams multiplied by 𝑣
𝑜. Then we can divide both sides
of the equation by 160 grams. So it cancels on the right. And on the left, we can see
that the unit of grams in the numerator cancels with the unit of grams in the
denominator. What this leaves us with is 𝑣
𝑜 on the right-hand side and some number with the unit centimeters per second
on the left-hand side.
Now this is perfect because
we’ve been asked to find the constant speed in centimeters per second that the
160-gram mass ball — that’s the orange ball — moves at. And what we’re finding here is
the speed with which the ball moves. We might think that we found
the velocity. But we actually know that the
velocity consists of the speed, which is what we’re calculating here, and the
direction, which we know is to the right. So the calculation is simply
going to give us the magnitude of the velocity, 𝑣 𝑜, or in other words the
speed with which the orange ball is moving.
Therefore, evaluating
everything on the left-hand side of the equation, we find that this is equal to
85 centimeters per second. And so we have our final
answer. The 160-gram mass ball after
the collision moves at a constant speed of 85 centimeters per second.
