Let's break down the problem to find the answers to each question. The problem involves a ball that is dropped from a height of $11$ feet, and each bounce reaches $\frac{6}{7}$ of the height of the previous bounce. We need to calculate:

1. The total vertical distance covered during the downward movement.
2. The total vertical distance covered during the upward movement.
3. The total vertical distance covered by the ball.

1. Distance in Descent

The ball initially falls from a height of $11$ feet. After each bounce, it reaches $\frac{6}{7}$ of the previous height. The distances for each descent form a geometric series with the first term $a = 11$ and common ratio $r = \frac{6}{7}$.

The total distance in descent is: $S_{\text{down}} = a + a \cdot r^2 + a \cdot r^4 + \dots$

Using the formula for the sum of an infinite geometric series, where $S = \frac{a}{1 - r}$, we get: $S_{\text{down}} = \frac{11}{1 - \left(\frac{6}{7}\right)^2}$

Calculating $\left(\frac{6}{7}\right)^2 = \frac{36}{49}$, so: $S_{\text{down}} = \frac{11}{1 - \frac{36}{49}} = \frac{11}{\frac{13}{49}} = 11 \cdot \frac{49}{13} = \frac{539}{13} \approx 41.46 \text{ feet}$

2. Distance in Ascent

The ascent distance also follows a geometric series, starting from the first bounce upward. Here, the first ascent is $\frac{6}{7} \cdot 11$ feet, and each subsequent ascent is $\frac{6}{7}$ of the previous one.

Thus, the ascent distance $S_{\text{up}}$ is: $S_{\text{up}} = \frac{\left(\frac{6}{7}\right) \cdot 11}{1 - \frac{6}{7}} = \frac{\frac{66}{7}}{\frac{1}{7}} = \frac{66}{7} \times 7 = 33 \text{ feet}$

3. Total Vertical Distance

The total vertical distance covered by the ball is the sum of the distances in descent and ascent: $S_{\text{total}} = S_{\text{down}} + S_{\text{up}} = \frac{539}{13} + 33 \approx 41.46 + 33 = 74.46 \text{ feet}$

Answers

• Distance in descent: Approximately $41.46$ feet.
• Distance in ascent: $33$ feet.
• Total vertical distance: Approximately $74.46$ feet.

Would you like additional details on these calculations?

Here are some additional questions to explore this topic:

1. How would the total distance change if the bounce ratio was different, say $\frac{5}{6}$?
2. How does the convergence of a geometric series work, and why does it apply to this bouncing problem?
3. What if the ball started with a different initial height, such as 15 feet? How would that affect the total distance?
4. Could we use calculus to approximate the total distance if it weren't an infinite series?
5. How does the energy loss affect the height of each bounce in physical terms?

Tip: For any sequence that involves repeated multiplications, check if it forms a geometric series, which can be summed up using the geometric series formula.