Suppose a particle of known mass and velocity collides with a second particle of known mass that is at rest. You try to solve for the final velocities of both particles in the plane of collision. How many unknowns could you possibly have to solve this problem

You could get by with 2 unknowns.

One final velocity and the character of the collision if the other final velocity is known

Both final velocities if the character of the collision is known.

Explanation:

You already have two known masses and both initial velocities.

The unknowns are two final velocities. Related to the two initial and two final velocities is the character of the collision.

We can already bracket the answers with the given information. The lower energy result being a perfectly in-elastic collision where the final velocity v is common to both and would be equal to the velocity of the Center of Mass of the system

m₁u₁ + m₂(0) = (m₁ + m₂)v

v = m₁u₁ / (m₁ + m₂)

The other end of the bracket would be the upper energy result from a perfectly elastic collision.  An elastic collision will have the relative velocity of approach equal the relative velocity of departure.

For approach the relative velocity is |u - 0| = u

For departure the relative velocity would be |v₂ - v₁| which will also = u.

Having one final velocity known is sufficient to solve the entire problem in either case.

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