\begin{tabular}{l}7 A bouncy rubber ball dropped from any height will lose \( 25 \% \) on its nergni on \\ each successive bounce. If dropped from a height of 30 m , the first bounce will \\ have a height of \( 22,5 \mathrm{~m} \) and the second bounce a height of \( 16,875 \mathrm{~m} \). Determine \\ the total vertical distance travelled by the ball from the time it is dropped from \\ a height of 40 m , until it finally comes to rest. Note that the height from which \\ the ball is dropped is 40 m , and not 30 m . \\ \hline\end{tabular}

To determine the total vertical distance traveled by the ball dropped from a height of \( 40 \) meters until it comes to rest, we'll analyze each phase of its motion. ### Understanding the Problem 1. **Initial Drop:** - The ball is initially dropped from a height of \( 40 \) meters. 2. **Successive Bounces:** - On each bounce, the ball retains \( 75\% \) of its previous height (since it loses \( 25\% \) of its energy). - This means the height after each bounce is multiplied by \( 0.75 \). 3. **Total Distance Calculation:** - The ball travels **downward** on the initial drop. - For each subsequent bounce, it travels **upward** and then **downward**. ### Step-by-Step Calculation 1. **Initial Drop:** \[ \text{Distance} = 40 \, \text{m} \] 2. **First Bounce:** \[ \text{Height after first bounce} = 40 \times 0.75 = 30 \, \text{m} \] \[ \text{Distance} = 30 \, \text{m (up)} + 30 \, \text{m (down)} = 60 \, \text{m} \] 3. **Second Bounce:** \[ \text{Height after second bounce} = 30 \times 0.75 = 22.5 \, \text{m} \] \[ \text{Distance} = 22.5 \, \text{m (up)} + 22.5 \, \text{m (down)} = 45 \, \text{m} \] 4. **Continuing Indefinitely:** - This pattern continues indefinitely, with each bounce contributing \( 2 \times \text{current height} \) to the total distance. - The heights form a geometric series: \[ 40 + 2(30 + 22.5 + 16.875 + \dots) \] 5. **Summing the Series:** - The heights form an infinite geometric series with: \[ \text{First term} (a) = 30 \, \text{m} \] \[ \text{Common ratio} (r) = 0.75 \] - The sum \( S \) of an infinite geometric series is: \[ S = \frac{a}{1 - r} = \frac{30}{1 - 0.75} = \frac{30}{0.25} = 120 \, \text{m} \] - Since each bounce contributes twice the height (up and down), the total distance from bounces is: \[ 2 \times 120 = 240 \, \text{m} \] 6. **Total Vertical Distance:** \[ \text{Initial drop} + \text{Bounces} = 40 + 240 = 280 \, \text{m} \] ### Conclusion The **total vertical distance** traveled by the ball from the time it is dropped until it comes to rest is: \[ \boxed{280 \text{ meters}} \]