### Video Transcript

Two spheres are moving along a
straight line. One has mass 𝑚 and is moving at
speed 𝑣, whereas the other has a mass of 10 grams 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.

There are two scenarios in this
question, and in both cases, the bodies coalesce. This means that they join together
and the collision is inelastic. In the first scenario, the two
bodies are moving in the same direction. In this situation, they join
together and move at a speed of 30 centimeters per second. In the second scenario, they were
originally moving towards each other. And they end up moving in the same
direction as the first sphere with a speed of six centimeters per second. In order to answer this question,
we will use the conservation of momentum. This states that the momentum
before is equal to the momentum after. The formula we will use is 𝑚 one
𝑢 one plus 𝑚 two 𝑢 two is equal to 𝑚 one 𝑣 one plus 𝑚 two 𝑣 two.

We will now clear some space to
answer the question. As the spheres are joining
together, we will only have one product on the right-hand side. In our first scenario, we have 𝑚𝑣
plus 10 multiplied by 36 is equal to 𝑚 plus 10 multiplied by 30. Distributing the parentheses gives
us 30𝑚 plus 300. We can then subtract 360 from both
sides so that 𝑚𝑣 is equal to 30𝑚 minus 60. We repeat this for the second
scenario. The only differences are the 36 is
now negative as the second sphere is moving in the opposite direction and the final
velocity is six. This simplifies to 𝑚𝑣 minus 360
is equal to six 𝑚 plus 60. This time, we can add 360 to both
sides, such that 𝑚𝑣 is equal to six 𝑚 plus 420.

We now have a pair of simultaneous
equations where the left-hand side of both of them is 𝑚𝑣. This means that 30𝑚 minus 60 must
be equal to six 𝑚 plus 420. Adding 60 and subtracting six 𝑚
from both sides gives us 24𝑚 is equal to 480. We can then divide both sides by 24
giving us 𝑚 is equal to 20. The mass of the first sphere is 20
grams. We can now substitute this into one
of our equations. We will choose equation one. This gives us 20𝑣 is equal to 30
multiplied by 20 minus 60. The right-hand side simplifies to
540. We can then divide both sides of
the equation by 20 giving us 𝑣 is equal to 27. The initial velocity of the first
sphere is 27 centimeters per second. We have now found the mass and
speed of the first sphere that would result in the two spheres coalescing on
collision.