After striking the floor, a rubber ball rebounds to 4/5th of the height from which it has fallen. Find the total distance that it travels before coming to rest if it has been gently dropped from a height of 120 metres.

Options

A. 540 metres
B. 960 metres
C. 1080 metres
D. 1020 metres

Answer

To find the total distance that the rubber ball travels before coming to rest, we can break down the problem step by step.

### Step 1: Initial Drop
The ball is dropped from a height of 120 meters. This is the first distance it travels downwards.

**Distance traveled downwards = 120 meters**

### Step 2: First Rebound
After hitting the ground, the ball rebounds to \( \frac{4}{5} \) of the height it fell from. Therefore, the height of the first rebound is:

\[ \text{Height of first rebound} = 120 \times \frac{4}{5} = 96 \text{ meters} \]

### Step 3: First Descent After Rebound
After reaching the height of 96 meters, the ball falls back down the same distance.

**Distance traveled downwards = 96 meters**

### Step 4: Second Rebound
The ball rebounds again to \( \frac{4}{5} \) of the height it just fell from (96 meters):

\[ \text{Height of second rebound} = 96 \times \frac{4}{5} = 76.8 \text{ meters} \]

### Step 5: Second Descent After Rebound
The ball falls back down from the height of 76.8 meters.

**Distance traveled downwards = 76.8 meters**

### Step 6: Continuing the Process
This process continues indefinitely, where each subsequent rebound height is \( \frac{4}{5} \) of the previous height. The distances traveled can be represented as a geometric series.

### Step 7: Total Distance Calculation
The total distance traveled by the ball can be calculated as follows:

1. **Total distance for downward travel**:
- First drop: 120 meters
- Subsequent drops: 96 + 76.8 + ... (which forms a geometric series)

2. **Total distance for upward travel**:
- First rebound: 96 meters
- Subsequent rebounds: 76.8 + ... (which also forms a geometric series)

### Step 8: Summing the Geometric Series
The downward distances after the first drop form a geometric series:
- First term \( a = 96 \)
- Common ratio \( r = \frac{4}{5} \)

The sum \( S \) of an infinite geometric series is given by:

\[ S = \frac{a}{1 - r} \]

Calculating the sum of the downward distances:

\[ S_{\text{down}} = 96 \times \frac{1}{1 - \frac{4}{5}} = 96 \times 5 = 480 \text{ meters} \]

The total downward distance is:

\[ \text{Total downward distance} = 120 + 480 = 600 \text{ meters} \]

The upward distances also form a geometric series with the same first term and common ratio:

\[ S_{\text{up}} = 96 \times \frac{1}{1 - \frac{4}{5}} = 480 \text{ meters} \]

### Step 9: Final Total Distance
The total distance traveled by the ball is the sum of the total downward and upward distances:

\[ \text{Total distance} = 600 + 480 = 1080 \text{ meters} \]

### Final Answer
The total distance that the ball travels before coming to rest is **1080 meters**.

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