Energy Conservation

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 is the ratio of the relative speed after collision to the relative speed before collision: For a ball falling on the floor, each time the ball hits the floor, its velocity decreases by a factor of (since the velocity after collision is times the velocity before collision).
Step 2: Distance Covered in Each Drop When the ball falls from height , it hits the floor and bounces back. The distance covered in the first fall is . After the first bounce, the ball reaches a height because its velocity after the bounce is reduced by a factor of , and the height is proportional to the square of the velocity. Thus, after the first bounce, the ball falls from height , then bounces back to , 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: This is a geometric series with the first term and common ratio . Using the formula for the sum of an infinite geometric series , where is the first term and is the common ratio, the total distance covered is: Simplifying the expression: Thus, the correct total distance covered by the ball before it comes to rest is:
Step 4: Conclusion The correct answer is: