A ball is dropped from a height of 10 m. After each bounce, the ball bounces up to 80% of its previous height. Determine the total distance travelled by the ball. Bouncing Ball GeoGebra.
Sketch a picture of the situation. The sum of the distances is,
\begin{gather*}
= 10 + 2 \cdot 10 \cdot 0.8 + 2 \cdot 10 \cdot 0.8^2 + 2 \cdot 10 \cdot 0.8^3 + \dots
\end{gather*}
It is better to write out the distances without doing the arithmetic, so that it is easier to recognize the pattern. This is a geometric series, except for the first number 10. Each next term is the same as the previous term, except multiplied by \(0.8\text{.}\) The first term of \(2 \cdot 10 \cdot 0.8 = 16\text{,}\) and the common ratio is \(r = 0.8\text{,}\)
\begin{gather*}
= 10 + \underbrace{2 \cdot 10 \cdot 0.8 + 2 \cdot 10 \cdot 0.8^2 + 2 \cdot 10 \cdot 0.8^3 + \dots}_{\text{geometric series, } a = 16, r = 0.8}
\end{gather*}
Then, using the infinite geometric series formula,
\begin{align*}
\amp = 10 + \frac{16}{1 - 0.8}\\
\amp = 90 \text{ m}
\end{align*}
Therefore, the total distance travelled is 90 m.
