Question

A ball falls from a height \( h \) upon a fixed horizontal floor. The coefficient of restitution between the ball and the floor is \( e \). The total distance covered by the ball before it comes to rest is:

• A. \( \frac{1 - e^2}{1 + e^2} h \)
• B. \( \frac{1 + e^2}{1 - e^2} h \)
• C. \( \frac{1 - 2e^2}{1 + e^2} h \)
• D. \( \frac{1 + 2e^2}{1 - e^2} h \)

Hint:

When dealing with collisions, the total distance covered by a bouncing object can often be found using the sum of an infinite geometric series. The coefficient of restitution helps determine the ratio of successive bounces.

Answer 1

The Correct Option is B

To solve this problem, we need to apply the concept of the coefficient of restitution and energy conservation.
Step 1: Understanding the Coefficient of Restitution The coefficient of restitution \( e \) is the ratio of the relative speed after collision to the relative speed before collision: \[ e = \frac{\text{velocity after collision}}{\text{velocity before collision}} \] For a ball falling on the floor, each time the ball hits the floor, its velocity decreases by a factor of \( e \) (since the velocity after collision is \( e \) times the velocity before collision).
Step 2: Distance Covered in Each Drop When the ball falls from height \( h \), it hits the floor and bounces back. The distance covered in the first fall is \( h \). After the first bounce, the ball reaches a height \( e^2 h \) because its velocity after the bounce is reduced by a factor of \( e \), and the height is proportional to the square of the velocity. Thus, after the first bounce, the ball falls from height \( e^2 h \), then bounces back to \( e^4 h \), and so on. Each successive fall and bounce will cover a smaller and smaller distance.
Step 3: Total Distance Covered The total distance covered by the ball is the sum of all the drops and bounces. This can be expressed as a series: \[ \text{Total Distance} = h + 2(e^2 h + e^4 h + e^6 h + \dots) \] This is a geometric series with the first term \( e^2 h \) and common ratio \( e^2 \). Using the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio, the total distance covered is: \[ \text{Total Distance} = h + 2 \cdot \frac{e^2 h}{1 - e^2} \] Simplifying the expression: \[ \text{Total Distance} = h + \frac{2e^2 h}{1 - e^2} = \frac{h(1 + e^2)}{1 - e^2} \] Thus, the correct total distance covered by the ball before it comes to rest is: \[ \boxed{\frac{1 + e^2}{1 - e^2} h} \]
Step 4: Conclusion The correct answer is: \[ \boxed{(B)} \]