Introduction:
When a ball is dropped from a certain height and bounces off the floor, it loses some of its initial energy due to the impact and rebound. In this case, the ball can only attain 4/5th of its previous height after each bounce.

Given:
- Initial height (h) = 1 m
- Height after each bounce = 4h/5

Approach:
To find the time it takes for the ball to come to rest, we need to calculate the total distance traveled by the ball.

1. Distance traveled during the first fall:
- The ball is initially dropped from a height of 1 m.
- The distance traveled during free fall can be calculated using the formula:
- s = ut + (1/2)gt^2, where u = initial velocity (0), g = acceleration due to gravity (-9.8 m/s^2), and t = time taken.
- Since the ball is being dropped, the initial velocity (u) is 0.
- We need to find the time taken for the ball to reach the floor, so we can rearrange the formula to:
- t = sqrt(2s/g), where s = initial height = 1 m.

2. Distance traveled during the first bounce:
- After the ball hits the floor, it will rebound to a height of 4h/5.
- The distance traveled during the bounce can be calculated using the formula:
- s = ut + (1/2)gt^2, where u = initial velocity (0), g = acceleration due to gravity (-9.8 m/s^2), and t = time taken.
- Since the ball is rebounding upwards, the final velocity (v) at the highest point is 0.
- We need to find the time taken for the ball to reach the highest point, so we can rearrange the formula to:
- t = -v/g, where v = final velocity = 0.

3. Total distance traveled:
- The total distance traveled by the ball can be calculated by summing the distances during the fall and each bounce.
- The ball travels the initial height (1 m) during the first fall, and then travels (4h/5) during each subsequent bounce.
- The total distance can be calculated using the formula:
- Total distance = 1 + 2 * (4h/5) + 2 * (4h/5)^2 + 2 * (4h/5)^3 + ...

4. Time taken to come to rest:
- The ball comes to rest when it no longer has sufficient height to bounce back.
- In this case, the height after each bounce is 4h/5.
- The ball will come to rest when the height after each bounce becomes negligible.
- We can consider the ball to have come to rest when the height after each bounce is less than the precision of our measurement.
- In this case, we can assume the height after each bounce is negligible when it is less than 0.01 m.

Calculation:
- Substituting the given values into the formula for total distance:
- Total distance = 1 + 2 * (4/5) + 2 * (4