# The Total Vertical Distance of a Ball Dropping from 10 Feet

• apiwowar
In summary, a ball dropped from a height of 10 feet bounces with each bounce being 3/4 of the previous height. The expression for the height hn after the nth bounce is 10(3/4)^n. To find the vertical distance Di traveled when the ball hits the floor for the first, second, third, and fourth times, the expression is 10 + 10(3/4)^n, taking into account that the ball goes up and down. To find the total vertical distance the ball has traveled when it hits the floor for the nth time, the geometric series must be summed.

#### apiwowar

a ball is dropped from a height of 10 feet, each bounce is 3/4 of the height of the bounce before

a)find an expression for the height hn to which the ball rises after it hits the floor for the nth time

so hn= 10(3/4)n

b) find an expression for the vertical distance Di the ball has traveled when it hits the floor for the first, second, third, and fourth times.

im a little confused on this part, how would Di relate to hh? or do they even relate at all?

would it be 10 + 10(3/4)n since after it hits the ground the first time it will decrease by 3/4 of the previous height each time?

c) find an expression for the total vertical distance the ball has traveled when it hits the floor for the nth time

im sure once figure out b this will make perfect sense

any hints or tips on how to do this would really be appreciated.

The first time it hits the floor it's traveled 10*(3/4)^0=10 feet. The second time it's traveled 10*(3/4)^0+2*10*(3/4)^1. The '2' is because it goes up and down. The third, 10*(3/4)^0+2*10*(3/4)^1+2*10*(3/4)^2. Etc. For the first few times you can just add it up by hand. For the nth time you have to sum the geometric series.