Video Transcript
Two spheres are moving along a
straight line. One has a mass of 𝑚 and is
moving at speed 𝑣, whereas the other one has a mass of 10 g and is moving at 36
centimeters per second. If the two spheres were moving
in the same direction when they collided, they would coalesce into one body and
move at 30 centimeters per second in the same direction. However, if they were moving in
opposite directions, they would coalesce into one body, which would move at six
centimeters per second in the direction the first sphere had been traveling. Find 𝑚 and 𝑣.
With the information we’ve been
given, we’re going to set up a pair of simultaneous equations in 𝑚 and 𝑣 using
conservation of momentum. In particular, we’re going to
use the formula momentum is equal to mass times velocity.
Let’s begin by identifying the
direction to the right to be positive. Then, we can say the momentum
of the first sphere is 𝑚 times 𝑣. The momentum of the second
sphere is 10 times 36. That’s 360, giving us a total
or net momentum before the collision of 𝑚𝑣 plus 360. By the conservation of
momentum, we know this must be equal to the net momentum after the
collision. The new mass of the body is 10
plus 𝑚, whilst its velocity is 30. Since it’s in the same
direction, it’s still positive, meaning the momentum after the collision is 30
times 10 plus 𝑚. By distributing parentheses and
rearranging, we can form an equation, 60 equals 30𝑚 minus 𝑚𝑣.
Now that we have this equation,
let’s complete the process for the second scenario. In this scenario, these spheres
are moving in opposite directions. This means the velocity of the
second sphere must be negative 36. So the net momentum is 𝑚𝑣
minus 360 before the collision. After the collision, they
coalesce, giving us a total mass of 10 plus 𝑚. And they move at six
centimeters per second, still in the positive direction. This time, if we distribute the
parentheses and rearrange, we get the equation negative 420 equals six 𝑚 minus
𝑚𝑣.
And now we might observe that
we can quite quickly calculate the value of 𝑚 here. We have a system of equations
in which we have the same term negative 𝑚𝑣. Let’s define these equations to
be one and two, respectively. Then, if we subtract one
equation from the other, and it doesn’t matter which way round we do it, that
negative 𝑚𝑣 term will disappear. 60 minus negative 420 is
480. Then, 30𝑚 minus six 𝑚 is
24𝑚, and negative 𝑚𝑣 minus negative 𝑚𝑣 is zero. If we then divide through by
24, we find that 𝑚 is equal to 20 or 20 grams.
Then, we can find the value of
𝑣 by substituting it into either of our earlier equations. When we do, we get 60 equals 30
times 20 minus 20𝑣, which gives us negative 540 equals negative 20𝑣. Finally, we divide through by
negative 20, and we find 𝑣 is equal to 27 centimeters per second. 𝑚 is 20 grams and 𝑣 is 27
centimeters per second.
