8. In the billiard ball room at the basement of Memorial Union, two identical balls are traveling toward each other along a straight line which is chosen to the x-axis. They experience an elastic head-on collision. Ignoring any frictional forces, if one ball m1 moving at a velocity of +4.67 m/s and the other m2 at a velocity of -7.89 m/s, what are the velocity (magnitude and direction) of each ball after the collision?

- Ball 1 will move backward at 7.89 m/s

- Ball 2 will move forward at 4.67 m/s

Explanation:

In an elastic collision, both the total momentum and the total kinetic energy of the system are conserved.

The conservation of the momentum can be written as:

where

is the mass of ball 1

is the mass of ball 2

is the initial  velocity of ball 1

is the initial velocity of ball 2

is the final velocity of ball 1

is the final velocity of ball 2

The conservation of kinetic energy can be written as

Working together the two equations, it is possible to find two expressions for the final velocities in terms of the initial velocities:

In this problem we have:

since the mass of the two balls is identical

is the initial velocity of ball 1

is the initial velocity of ball 2

Substituting into the equations, we find the final velocities:

Therefore:

- Ball 1 will move backward at 7.89 m/s

- Ball 2 will move forward at 4.67 m/s

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