Geometric Series: Bouncing Ball — a Ball Bounces

Does an infinite geometric series apply to a bouncing ball?
Refer to the beginning of chapter of sequences and series. Suppose you wrote a geometric series to find the sum of the heights of the rebounds of the ball. The series would have no last term because theoretically there is no last bounce of the ball. For every rebound of the ball, there is another rebound, ⅔ as high. Such a geometric series is called an infinite geometric series.

(How). Do geometric sequences apply to a bouncing ball?

Question 1. (Physical Science) Maddy drops a ball off of a building that is 60 feet high. Each time the ball bounces, it bounces back to ⅔ its previous height. If the ball continues to follow this pattern what will be the total distance that the ball travels?
Solution:
Substitute 60 for a1 and ⅔ for r in the sum formula.


Therefore, the total distance traveled by the ball is 2⋅180-60 or 300 ft.
Answer: 300 ft

Example 1. (Challenge) A ball is dropped from a height of 5 meters. On each bounce, the ball rises to 65% of the height it reached on the previous bounce.

ball bounces

a. Approximate the total vertical distance the ball travels, until it stops bouncing.
b. The ball makes its first complete bounce in 2 seconds, that is, from the moment it first touches the ground until it next touches the ground. Each complete bounce that follows takes 0.8 times as long as the preceding bounce. Estimate the total amount of time that the ball bounces.
Solution:
a. The ball is dropped from a height of 5 meters, bounces back up 0.65(5) or 3.25 meters, falls 3.25 meters, bounces back up 0.65(3.25) or 2.1125 meters, falls 2.1125 meters, and so on. So, an infinite sequence that can be used to represent this situation is 5, 3.25, 3.25, 2.1125, 2.1125, … . The corresponding series can be written as the sum of the two infinite geometric series: one series that represents the distance the ball travels when falling and one series that represents the distance the ball travels when bouncing back up.
Series 1 5+3.25+2.1125+⋯
Series 2 3.25+2.1125+1.373125+⋯
Find the sum of each series.
Infinite geometric series formula

Therefore, the total vertical distance the ball travels is 14.29+9.29 or about 23.6 meters.
b. In this sequence, a1=2 and r=0.8. Find the sum of the related series.
Infinite geometric series formula

Therefore, the total amount of time that the ball bounces is 10 seconds.

Example 2. (EXPERIMENT) In a physics class experiment, Diana drops a ball from a height of 16 feet. Each bounce has 70% the height of the previous bounce. Draw a graph to represent the height of the ball after each bounce.
Solution:
Make a table of values.

Bounce Ball Height
1 0.7(16)=11.2
2 0.7(11.2)=7.84
3 0.7(7.84)=5.488
4 0.7(5.488)=3.8416
5 0.7(3.8416)=2.68912
6 0.7(2.68912)=1.882384
7 0.7(1.882384)=1.3176688

Graph the bounce on the x-axis and the ball height on the y-axis. Experiment

bounce graph

Example 3. If a ball has elasticity such that it bounces up 80% of its previous height, find the total vertical distances travelled down and up by this ball when it is dropped from an altitude of 3 metres. Ignore friction and air resistance.
Solution:

a ball bounces

After the ball is dropped the initial 3 m, it bounces up and down a distance of 2.4 m. Each bounce after the first bounce, the ball travels 0.8 times the previous height twice — once upwards and once downwards. So, the total vertical distance is given by
h=3+2(2.4+ (2.4×0.8)+(2.4×0.82)+…)=3+2×1
The amount in parenthesis is an infinite geometric series with a1=2.4 and r=0.8. The value of that quantity is

Hence, the total distance required is h=3+2(12)=27 m.

Example 4: Using an Infinite Series as a Model

ball bounce

BALL BOUNCE
This photo of a ball bouncing was taken with time-lapse photography. The images of the ball get closer together as you move up, which means the ball’s speed is decreasing.

A ball is dropped from a height of 10 feet. Each time it hits the ground, it bounces to 80% of its previous height.
a. Find the total distance traveled by the ball.
b. On which bounce will the ball have traveled 85% of its total distance?

bounce number

Solution:
a. The total distance traveled by the ball is:

(Excluding first term, find sum of series.)

(Simplify fraction.)
=10+80
(Simplify.)
=90.
► The basll travels a total distance of 90 feet.
b. Let n be the number of up-and—down bounces. The distance dn the ball travels is:
(Write rule for dn)

► The ball travels 85% of its total distance after about 8 up-and-down bounces, or after 9 bounces including the first down-only bounce.

Question 2. (Toys) If a rubber ball can bounce back to 95% of its original height, what is the total vertical distance that it will travel if it is dropped from an elevation of 30 feet?
Solution:
Distance traveled by the rubber ball in downward direction.
Given, a1=30 and r=95% or 0.95.
Find the sum.


Distance traveled by the rubber ball in upward direction.
Given, a1=28.5 and r=95% or 0.95.
Find the sum.

The total distance traveled by the rubber ball is 600+570 or 1170 ft.
Answer: 1170 ft

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