A ball bounces to 45 percent of its original height.
What fraction of its mechanical energy is lost each time it bounces?
What is the coefficient of restitution of the ball-floor system?
(a) 55% is lost and 45% is kept after each bounce.
(b) 0.45
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To find the fraction of mechanical energy lost each time the ball bounces, we need to calculate the ratio of the energy lost to the original energy.
1. First, note that mechanical energy is the sum of kinetic energy and potential energy. When the ball is at its highest point (the maximum height), it has purely potential energy. When it is at its lowest point (the ground), it has purely kinetic energy.
2. Since the ball bounces back up to 45 percent of its original height, we can conclude that it loses 55 percent of its original mechanical energy during each bounce. This is because it loses some energy due to internal damping and external factors like air resistance.
3. Therefore, the fraction of mechanical energy lost each time the ball bounces is 55 percent or 0.55.
To find the coefficient of restitution of the ball-floor system, we need to understand that it measures how well the ball bounces back after colliding with the floor. It is usually denoted by the symbol 'e.'
The coefficient of restitution can be calculated as the square root of the ratio of the rebound height to the drop height.
1. Assume the original drop height of the ball is 'h'. After bouncing, it reaches a height of 45 percent of 'h', which is 0.45h.
2. The coefficient of restitution (e) is then given by the square root of (0.45h / h), which simplifies to the square root of 0.45.
3. Using a calculator or mathematical software, find the square root of 0.45, which gives approximately 0.67.
Therefore, the coefficient of restitution of the ball-floor system in this case is approximately 0.67.