A ball is droped from a height 500 metre. every time ball is bounced back up to the height 4 / 5th of previous height, till the ball stopped, how much distance is covered by the ball ?

Options

A. 4500 m
B. 4000 m
C. 4800 m
D. 5000 m

Answer

To solve the problem of how much distance is covered by the ball dropped from a height of 500 meters, we can break it down step by step.

### Step-by-Step Solution:

1. **Initial Drop**: The ball is dropped from a height of 500 meters.
- Distance covered during the drop = 500 meters.

2. **First Bounce**: After hitting the ground, the ball bounces back to a height of \( \frac{4}{5} \) of the previous height (500 meters).
- Height after first bounce = \( 500 \times \frac{4}{5} = 400 \) meters.
- Distance covered during the bounce = 400 meters (upward).

3. **Second Drop**: The ball then falls back down from 400 meters to the ground.
- Distance covered during the drop = 400 meters (downward).

4. **Second Bounce**: The ball bounces back to a height of \( \frac{4}{5} \) of 400 meters.
- Height after second bounce = \( 400 \times \frac{4}{5} = 320 \) meters.
- Distance covered during the bounce = 320 meters (upward).

5. **Third Drop**: The ball falls back down from 320 meters to the ground.
- Distance covered during the drop = 320 meters (downward).

6. **Third Bounce**: The ball bounces back to a height of \( \frac{4}{5} \) of 320 meters.
- Height after third bounce = \( 320 \times \frac{4}{5} = 256 \) meters.
- Distance covered during the bounce = 256 meters (upward).

7. **Continuing the Process**: This process continues indefinitely, with the height after each bounce being \( \frac{4}{5} \) of the previous height.

8. **Total Distance Calculation**:
- The total distance covered by the ball can be calculated as:
\[ \text{Total Distance} = \text{Distance of drop} + \text{Distance of bounces} \]
- The distance of drops is a series:
\[ 500 + 400 + 320 + 256 + \ldots \]
- The distance of bounces is also a series:
\[ 400 + 320 + 256 + \ldots \]

9. **Using the Formula for Infinite Geometric Series**:
- The distance of the bounces forms a geometric series where:
- First term \( a = 400 \)
- Common ratio \( r = \frac{4}{5} \)
- The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
- Thus, the total distance of bounces:
\[ S = \frac{400}{1 - \frac{4}{5}} = \frac{400}{\frac{1}{5}} = 400 \times 5 = 2000 \text{ meters} \]

10. **Final Total Distance**:
- Total distance covered by the ball:
\[ \text{Total Distance} = 500 + 2000 = 2500 \text{ meters} \]
- However, we need to account for the distance of all the upward and downward movements, which doubles the distance of the bounces:
\[ \text{Total Distance} = 500 + 2 \times 2000 = 4500 \text{ meters} \]

### Conclusion:
The total distance covered by the ball until it stops is **4500 meters**.