Question

3. A ball tossed to a height of 4 meters rebounds to 40% of its previous height. Find the distance the ball has traveled when it strikes the ground for the fifth time.

Answer 1

To find the total distance traveled by the ball when it strikes the ground for the fifth time, we can analyze its motion as follows:

1. **First toss:** The ball is tossed to a height of 4 meters.
2. **Rebound to 40%:** After reaching the height of 4 meters, it rebounds to 40% of that height, which is:
\[
0.4 \times 4 = 1.6 \text{ meters}
\]

3. **Motion sequence:**
- The ball travels up 4 meters (when tossed).
- It then falls down 4 meters (to the ground).
- It rebounds up 1.6 meters.
- It falls down 1.6 meters.
- Then it will rebound to:
\[
0.4 \times 1.6 = 0.64 \text{ meters}
\]
- It falls down 0.64 meters.
- The next rebound height will be:
\[
0.4 \times 0.64 = 0.256 \text{ meters}
\]
- It falls down 0.256 meters.
- Finally, it will rebound to:
\[
0.4 \times 0.256 = 0.1024 \text{ meters}
\]
- It will fall down 0.1024 meters.

Now we can calculate the total distance as follows. We sum all the upward and downward distances traveled until the fifth impact with the ground.

### Total Distance Calculation

Now we list down all movements until the fifth strike:

1. **First Toss (Up):**
- Up: \(4 \text{ m}\)
- Down: \(4 \text{ m}\)

2. **Second Rebound:**
- Up: \(1.6 \text{ m}\)
- Down: \(1.6 \text{ m}\)

3. **Third Rebound:**
- Up: \(0.64 \text{ m}\)
- Down: \(0.64 \text{ m}\)

4. **Fourth Rebound:**
- Up: \(0.256 \text{ m}\)
- Down: \(0.256 \text{ m}\)

5. **Fifth Rebound: (Final Up)**
- Up: \(0.1024 \text{ m}\)
- The ball will strike the ground after it goes up \(0.1024 \text{ m}\) and comes down again, completing the motion for the fifth time.

### Adding Distances

Now summing all the distances:
\[
\text{Total Distance} = 4 + 4 + 1.6 + 1.6 + 0.64 + 0.64 + 0.256 + 0.256 + 0.1024 + 0.1024
\]

This can be calculated as follows:

- Initial Height: \(4 + 4 = 8\) meters.
- First Rebound: \(1.6 + 1.6 = 3.2\) meters.
- Second Rebound: \(0.64 + 0.64 = 1.28\) meters.
- Third Rebound: \(0.256 + 0.256 = 0.512\) meters.
- Fourth Rebound: \(0.1024 + 0.1024 = 0.2048\) meters.

Now adding all these distances together:
\[
\text{Total Distance} = 8 + 3.2 + 1.28 + 0.512 + 0.2048
\]
Calculating this step-by-step:
1. \(8 + 3.2 = 11.2\)
2. \(11.2 + 1.28 = 12.48\)
3. \(12.48 + 0.512 = 12.992\)
4. \(12.992 + 0.2048 = 13.1968\)

Thus, the total distance traveled by the ball when it strikes the ground for the fifth time is:
\[
\boxed{13.1968 \text{ meters}}
\]