Final answer:
The coefficient of restitution when 1/4th of the kinetic energy is lost in a collision is calculated as the square root of the remaining energy in terms of velocity. Since 3/4th of the energy remains, the coefficient is the square root of 3/4, or 0.5.
Step-by-step explanation:
The question asks to determine the coefficient of restitution when 1/4th of the kinetic energy is lost during the impact of a ball with a similar one at rest. The coefficient of restitution (c) is a dimensionless quantity that measures the elasticity of a collision, defined by the ratio of relative speeds after and before the impact. In an elastic collision, c equals 1, indicating no kinetic energy is lost. In contrast, an inelastic collision where energy is lost, has c less than 1.
To calculate c for a situation where 1/4th of the kinetic energy is lost, we know that kinetic energy is proportional to the square of the velocity. If 1/4th is lost, 3/4th remains. Taking the square root of 3/4th to relate it to velocity (since kinetic energy is proportional to the square of the velocity), we get that the ratio of the final to initial speed is √(3/4), which simplifies to 1/2 or 0.5. Therefore, the coefficient of restitution for this collision is 0.5.