# A ball is dropped from a height of 10 feet.Each time it hits the ground, it bounces to 80% of it's previous height. Find the total distance travelled by the ball?

Jun 10, 2018

See explanation.

#### Explanation:

Although the question is places in Physics, it can be answered using maths only.

The heights of the ball form a geometric sequence with ${h}_{1} = 10$ and the ratio $r = 0.8$. The ratio is less than $1$, so the sequence is convergent (i.e. it has a finite sum). The sum of the sequence can be calculated as:

## $S = {h}_{1} / \left(1 - r\right)$

$S = \frac{10}{1 - 0.8} = \frac{10}{0.2} = \frac{100}{2} = 50$

Jun 10, 2018

$\boldsymbol{= 90}$

#### Explanation:

$d = 10 + \textcolor{red}{2} \cdot 10 \cdot 0.8 + \textcolor{red}{2} \cdot 10 \cdot {0.8}^{2} + \ldots .$

$= 10 \left(1 + 2 \cdot 0.8 + 2 \cdot {0.8}^{2} + \ldots .\right)$

$= 10 \left(2 + 2 \cdot 0.8 + 2 \cdot {0.8}^{2} + \ldots . + 2 \cdot {0.8}^{k - 1} + \ldots\right) - 10$

$= 10 \cdot \frac{2}{1 - 0.8} - 10$

$\boldsymbol{= 90}$