A ball with a mass of #4 kg # and velocity of #1 m/s# collides with a second ball with a mass of #5 kg# and velocity of #- 8 m/s#. If #25%# of the kinetic energy is lost, what are the final velocities of the balls?

This is the answer of the velocity of the final ball (9 kg). It moves similar to the direction of the second ball with a lower speed.

To find the final velocities of the balls after the collision, we can use the principle of conservation of momentum and the equation for kinetic energy.

Let ( m_1 = 4 , \text{kg} ) be the mass of the first ball, ( v_1 = 1 , \text{m/s} ) its initial velocity, ( m_2 = 5 , \text{kg} ) the mass of the second ball, and ( v_2 = -8 , \text{m/s} ) its initial velocity.

Conservation of momentum: ( m_1 v_1 + m_2 v_2 = m_1 v_{1f} + m_2 v_{2f} )

Using the given values: ( (4 , \text{kg} \times 1 , \text{m/s}) + (5 , \text{kg} \times -8 , \text{m/s}) = (4 , \text{kg} \times v_{1f}) + (5 , \text{kg} \times v_{2f}) )

After solving this equation, we get the values of ( v_{1f} ) and ( v_{2f} ).

Now, to find the change in kinetic energy, we can use the formula:

( \text{Change in kinetic energy} = \text{Initial kinetic energy} - \text{Final kinetic energy} )

( \text{Initial kinetic energy} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 )

( \text{Final kinetic energy} = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 )

Given that 25% of the kinetic energy is lost, we can set up the equation:

( \text{Change in kinetic energy} = 0.25 \times \text{Initial kinetic energy} )

Substitute the expressions for initial and final kinetic energies into this equation and solve for the final velocities.