We will use geometric series to investigate the behavior of a bouncing ball. We are interested in the distance traveled and time bouncing.

Part A: How Far Does the Ball Travel?

We are going to use sequences and series to determine the behavior of a bouncing ball. Suppose you drop a ball from h feet above a hard, flat surface. Each time the ball bounces, it goes back up into the air by a factor of r times its previous height. The value of r depends on the elasticity of the ball and the physical properties of the surface. The phrase used for r is the coefficient of restitution. If the ball is a typical ball that we use for games (e.g., basketball, soccer ball, etc.), then r < 1. If the coefficient of restitution is larger or equal to 1, then the ball must have some other worldly properties.

Suppose, we drop the ball from 10 feet and see that it rebounds 0.51 feet on the first bounce. Therefore, h=10 and r=0.51. What is the height of the 6^th bounce? Since each bounce produces a height of r times the previous, the first bounce is hr and the second has height hr². The height of the 6^th bounce is hr⁶ For these values, we find the height of the 6^th bounce by evaluating 10 * (0.51)⁶ = 0.16 feet or about 2 inches.

How far does the ball travel during those 6 bounces? The distance traveled is made up of 3 parts:

• the first drop to the surface (10 feet for our ball)

• the 5 bounces up and back down (each bounce adds 2(10) (0.51)^k feet to distance for k = 1 to 5)
• the final (or 6^th) bounce upward: 10 (0.51)⁶ (the value we determined). If we put this all together in a summation, we find the total distance traveled.
• 
• Let’s compute this value.

Now enter the equation:

We need to multiply this by 2 and add the rest of it:

The total distance traveled is about 30 feet.

How far the does the ball travel after an infinite number of bounces. Our summation becomes an infinite series. If we put this all together we find the total distance traveled for a ball dropped from height h and rebounding by a factor r as:

What if we return to our example with h = 10 and r = 0.51? We Compute

Or about 31 feet. The infinity key is in green above the [catalogue] key on the 89 and in yellow above the [j] on the 92.

This is a geometric series with r < 1. The series sums to:
 
 

It is amazing that the ball bounces an infinite number of times but travels only a finite distance. How long it takes to complete these infinite number of bounces? By recalling the formula for the distance traveled for a dropped ball with respect to time t (motion under gravity), we model this motion as s(t)=16t,² where s is distance in feet and t is time in seconds (remember the acceleration due to gravity is g=32 ft/sec²). Solving for t, we obtain :

We can easily determine the time it takes for the first bounce if we start at h=10,

Knowing the height of each bounce, we can determine the time it takes each bounce (up and back down) to occur. Since the ball has to travel up and down, the distance traveled is twice the bounce height. Therefore, the time to perform the second bounce (up and down) with a bounce factor r, is:

Once again, we can sum these terms to get the amount of time to perform any number of bounces. If we go back to our first scenario in Part I, we have h=10, r=0.51, and number of bounces n=6. Our sum for the total time for these 6 bounces is:

or about 4 seconds!

How long does it take to bounce the infinite number of times? We convert this finite sum to an infinite series. Does this series converge? Is it a geometric series? What do you think? Let’s compute it,and see what happens. We just change 5 to infinity and simplify the radical to do this computation.

Not much longer!

Part 2: You Try it.

1. What distance does it travel after 10 bounces?

 

 
 
 
 
 
 
 

2. What distance does it travel after an infinite number of bounces?

 

 
 
 
 
 
 
 

3. What time does it take to make 10 bounces?

 

 
 
 
 
 
 
 

4. How long does it take to make an infinite number of bounces?

1. What distance does it travel after 10 bounces?

 

 
 
 
 
 
 
 

2. What distance does it travel after an infinite number of bounces?

 

 
 
 
 
 
 
 

3. What time does it take to make 10 bounces?

 

 
 
 
 
 
 
 

4. How long does it take to make an infinite number of bounces?