Video Transcript
Two spheres, 𝐴 and 𝐵, of
equal mass were projected toward each other along a horizontal straight line at
19 centimeters per second and 29 centimeters per second, respectively. As a result of the impact,
sphere 𝐵 rebounded at 10 centimeters per second. Find the velocity of sphere 𝐴
after the collision given that its initial direction is the positive
direction.
We first see that we’re told
that the initial direction of sphere 𝐴 is positive. So we might begin by drawing a
sketch of the scenario. We have sphere 𝐴 and 𝐵 moving
towards one another at 19 centimeters per second and 29 centimeters per second,
respectively. Sphere 𝐴 is moving in the
positive direction, which means sphere 𝐵 is moving in the negative
direction. We’re also told that each
sphere has an equal mass. So let’s define the mass of
each of our spheres to be 𝑚 grams.
Now, in practice, we tend to
work with kilograms and meters per second. But here we’re working with
centimeters per second. And so it’s more usual to be
consistent and use grams. We’re going to begin by
calculating the initial momentum of this system. We know that momentum is
conserved. So this will allow us to
compare the initial momentum with the final momentum, which should then in turn
allow us to calculate the final velocity of sphere 𝐴. We also know that we often
calculate momentum by using vector quantities for momentum and velocity. Since scalar momentum and
scalar velocity can be thought of as vectors in one direction, this means we can
generalize this for scalar quantities also.
So, we know that we can define
the initial momentum of sphere 𝐴 as 𝑝 sub 𝐴, 𝑖. And it’s its mass times
velocity. That’s 𝑚 times 19 or 19𝑚. In a similar way, we can
calculate the initial momentum for sphere 𝐵. 𝑝 sub 𝐵, 𝑖, that’s the
initial momentum, is mass times velocity. So that’s 𝑚 times negative 29
or negative 29𝑚. Then, we can calculate the
total momentum in the system — let’s call that 𝑝 sub net 𝑖, the initial net
momentum — by finding the sum of the initial momentum of 𝐴 and 𝐵. That’s 19𝑚 plus negative 29𝑚,
which is negative 10𝑚.
Now that we’ve identified
what’s happening before the collision, let’s think about what’s happening
immediately after. We’re told that the spheres
rebound. In other words, they move
immediately after the collision in opposite directions. We’re told sphere 𝐵 rebounds
at 10 centimeters per second, so it travels in the opposite direction at this
velocity. We’re trying to find the
velocity of sphere 𝐴. So let’s define that to be 𝑣
centimeters per second. And we’re assuming it’s moving
in the negative direction.
Then, we calculate the net
momentum by considering the final momentum of each sphere. The final momentum of sphere 𝐴
is going to be negative 𝑚𝑣, whilst the final momentum of sphere 𝐵, let’s call
that 𝑝 sub 𝐵, 𝑓, is mass times velocity, that’s 10𝑚. Then, we find their sum to find
the final net momentum. That’s negative 𝑚𝑣 plus
10𝑚.
Now, according to the principle
of conservation of momentum, the net momentum before the collision must be equal
to the net momentum after the collision. In other words, negative 10𝑚
must be equal to negative 𝑚𝑣 plus 10𝑚. Let’s clear a little bit of
space and solve this equation. To do so, we might first begin
by noticing that each expression in this equation contains a factor of 𝑚. We also defined 𝑚 to be the
mass of each object, so it cannot be equal to zero, meaning that we can divide
our entire equation by 𝑚. When we do, we get negative 10
equals negative 𝑣 plus 10. Subtracting 10 from both sides,
and our equation becomes negative 20 equals negative 𝑣.
And so the velocity of the
sphere after the collision is 20 centimeters per second. But of course, we define this
motion to be to the left. Since velocity is directional
and we define the direction to the right to be positive, we have to say that the
final velocity of sphere 𝐴 is negative 20 centimeters per second.
Now, at this stage, it’s worth
noting that we could have alternatively modeled the motion after the collision
slightly differently. In other words, we could have
defined the arrow 𝑣 centimeters per second to be moving to the right. If we’ve done that at this
stage, we would have ended up with 𝑣 equals negative 20 centimeters per second,
thereby still indicating that the object is moving to the left. Either way, the velocity is
negative 20 centimeters per second.
