A ball with a mass of #2# #kg # and velocity of #5# # ms^-1# collides with a second ball with a mass of #7# #kg# and velocity of #- 4# #ms^-1#. If #40%# of the kinetic energy is lost, what are the final velocities of the balls?

Momentum prior to impact:

Kinetic energy prior to impact:

Momentum following the impact:

kinetic energy following impact:

Replace in Equation 2:

This is a quadratic equation that can be resolved in your preferred way or by using the quadratic formula:

Equation 1's result after these values are substituted is:

Using the principle of conservation of momentum and the given information, we can find the final velocities of the balls.

Let ( m_1 = 2 , \text{kg} ) be the mass of the first ball, ( v_1 = 5 , \text{m/s} ) be its initial velocity, ( m_2 = 7 , \text{kg} ) be the mass of the second ball, and ( v_2 = -4 , \text{m/s} ) be its initial velocity.

The total initial momentum is ( p_{\text{initial}} = m_1v_1 + m_2v_2 ).

Using the principle of conservation of momentum, the total final momentum is equal to the initial momentum: ( p_{\text{final}} = p_{\text{initial}} ).

We can calculate the final velocities of the balls using the given information and the fact that kinetic energy is lost.

Let ( v_{1f} ) and ( v_{2f} ) be the final velocities of the first and second balls, respectively.

We also know that 40% of the kinetic energy is lost, so the final kinetic energy is ( 0.6 \times \text{initial kinetic energy} ).

Given that kinetic energy is ( \frac{1}{2}mv^2 ), we can write the equation for kinetic energy before and after the collision.

For ball 1: [ \frac{1}{2}m_1v_1^2 = \frac{1}{2}m_1{v_{1f}}^2 ]

For ball 2: [ \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_2{v_{2f}}^2 ]

We can solve these equations to find ( v_{1f} ) and ( v_{2f} ).