Final answer:
The fraction of kinetic energy lost in a collision with a coefficient of restitution of 0.69 is approximately 52.39%. This is calculated using the formula (1 - e^2), where e is the coefficient of restitution. The calculation reveals that the kinetic energy loss is more than half of the initial kinetic energy. The fraction of kinetic energy lost during a single collision with a coefficient of restitution of 0.69 is approximately 52.39%, not 21% as was initially supposed.
Step-by-step explanation:
The coefficient of restitution is a measure of how the speed of an object changes after a collision, indicating how much energy is lost. For a coefficient of restitution of 0.69, we calculate the fraction of kinetic energy lost during a collision as follows:
- First, note that the coefficient of restitution (e) is the ratio of the final speed (v') to the initial speed (v) of an object after it bounces: e = v' / v.
- Since kinetic energy (KE) is proportional to the square of the speed (KE ≈ v^2), we can find the initial and final kinetic energies as KE_initial = 0.5 * m * v^2 and KE_final = 0.5 * m * v'^2, where m is the mass of the object.
- The fraction of kinetic energy lost is then (KE_initial - KE_final) / KE_initial, which simplifies to (v^2 - v'^2) / v^2 after canceling out the mass (since it's the same before and after the collision).
- Substitute the coefficient of restitution for v'/v: (1 - e^2).
- Plug in the given coefficient of restitution (0.69) to get the fraction of energy lost: (1 - 0.69^2) = (1 - 0.4761) = 0.5239 or 52.39%.
The fraction of kinetic energy lost during a single collision with a coefficient of restitution of 0.69 is approximately 52.39%, not 21% as was initially supposed.