Thanks for any help
A ball drops from a height of 13 feet. Each time it hits the ground, it bounces up 70 percents of the height it fall. Assume it goes on forever, find the total distance it travels.
Hello
Let's call $\displaystyle d_k$ the distance (in feet) that the ball travels between the $\displaystyle k$-th and $\displaystyle (k+1)$-th bounce.
What is $\displaystyle d_0$ ? $\displaystyle d_1$ ? $\displaystyle d_2$ ? $\displaystyle d_{k+1}$ in function of $\displaystyle d_k$ ? $\displaystyle d_k$ in function of $\displaystyle k$ ?
This is a classic geometric series.
Each is 70% of the previous.
Downward the ball travels 13+9.1+6.37+4.459+.........
Upward the ball travels 9.1+6.37+4.459+..............
We can assume the total distance the ball travels, S, can be found by adding the infinite series.
$\displaystyle S=13+2(9.1+6.37+4.459+...........)$
$\displaystyle =13+2\left[9.1+9.1(.7)+9.1(.7)^{2}+9.1(.7)^{3}+..........\rig ht]$
Now we can use the formula $\displaystyle S=\frac{a_{1}}{1-r}$
Where $\displaystyle a_{1}=9.1, \;\ r=.7$
$\displaystyle S=13+2\left[\frac{9.1}{1-.7}\right]=73.6667$