A rubber ball rebounds 3/5 of the height from which it falls. If it is initially dropped from a height of 30 ft. What total vertical distance does it travel before coming to rest?

Make a diagram

distance during first bounce - 15 ft
distance during 2nd bounce - 2(15)(3/5)
distance during 3rd bounce = 2(15)(3/5)^2
distance during 4th bounce = 2(15)(3/5)^3
etc

from the 2nd term on, we have a geometric series
where a = 2(15)(3/5) = 18 and r = 3/5
and the want Sum∞
sum∞ = 15 + a/(1-r)
= 15 + 18/(1-3/5)
= 15 + 18/(2/5)
= 15 + 45 = 60

a sub 1 = 30

r = 3/5

S = (r) / (1-r)
S = {(30ft) / [(1)-(3/5)]}(3/5)
S = 150ft / 5-3 = 150ft / 2 = 70ft

60

120 ft

120...

30+ summation(X to infinity)(30*2*(3/5)^x)
30+90=120

S = 30 + 2 [15/(1-3/5)]

S = 30 + 2(15/2/5)
S = 30 +75
S = 105 ft

To find the total vertical distance traveled by the rubber ball before coming to rest, we need to calculate the sum of the distances it covers during each rebound.

The ball rebounds 3/5 of the height from which it falls, meaning it reaches 3/5 of its previous height with each bounce. Let's break down the problem into smaller steps:

1. Calculate the distance traveled by the ball during the first bounce:
- The ball is initially dropped from a height of 30 ft.
- On the first bounce, it reaches 3/5 * 30 ft = 18 ft.
- Therefore, during the first bounce, the ball covers a distance of 30 ft (drop) + 18 ft (rebound) = 48 ft.

2. Calculate the distance traveled by the ball during the second bounce:
- After the first bounce, the ball reaches a height of 18 ft.
- On the second bounce, it reaches 3/5 * 18 ft = 10.8 ft.
- Therefore, during the second bounce, the ball covers a distance of 18 ft (drop) + 10.8 ft (rebound) = 28.8 ft.

3. Calculate the distance traveled by the ball during the third bounce:
- After the second bounce, the ball reaches a height of 10.8 ft.
- On the third bounce, it reaches 3/5 * 10.8 ft = 6.48 ft.
- Therefore, during the third bounce, the ball covers a distance of 10.8 ft (drop) + 6.48 ft (rebound) = 17.28 ft.

We can continue this pattern and sum up the distances until the ball comes to rest. Alternatively, we can use the formula for the sum of an infinite geometric series:

S = a / (1 - r),

where:
S = sum of the series (total distance traveled),
a = first term of the series (30 ft),
r = common ratio (3/5).

Substituting the values into the formula, we get:

S = 30 / (1 - 3/5) = 30 / (2/5) = 30 * (5/2) = 75 ft.

Therefore, the rubber ball travels a total vertical distance of 75 ft before coming to rest.