A ball is dropped from a height of 30 feet. Each time it...

A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height. (a) What height will the ball bounce up to after it strikes the ground for the third time? (b) How high will it bounce after it strikes the ground for the $n$ th time? (c) How many times does the ball need to strike the ground before its bounce is less than 6 inches? (d) What total vertical distance does the ball travel before it stops bouncing?

A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height.
(a) What height will the ball bounce up to after it strikes the ground for the third time?
(b) How high will it bounce after it strikes the ground for the $n$ th time?
(c) How many times does the ball need to strike the ground before its bounce is less than 6 inches?
(d) What total vertical distance does the ball travel before it stops bouncing?

Key Concepts

Infinite Geometric Series

An infinite geometric series is the sum of all terms in a geometric sequence when the sequence continues indefinitely, particularly when the absolute value of the common ratio is less than one. This concept is used to calculate the total distance traveled by the bouncing ball as the sum of the distances of all bounces, which converges to a finite value despite an infinite number of bounces.

Logarithms

Logarithms are the inverse operations of exponentiation and are used to solve for the exponent in equations involving exponential growth or decay. They are crucial when determining the number of bounces required for the rebound height to drop below a certain threshold.

Geometric Sequences

Geometric sequences are ordered sets of numbers where each term after the first is found by multiplying the previous term by a constant value called the common ratio. In the context of bouncing balls, the height after each bounce forms a geometric sequence because the height reached after each bounce is a fixed fraction of the height from which it fell.

Exponential Decay

Exponential decay describes processes in which a quantity decreases at a rate proportional to its current value, which is a common phenomenon in natural systems. In the bouncing ball problem, the rebound height decreases by a constant fraction every time, illustrating exponential decay as the ball continues to bounce.

Transcript

00:01 We're describing the bouncing motion of a ball, which after every bounce rises to just 0 .8 of the height that it was previously at, that is, 80 % of its previous height.

00:10 Its initial height was 30 feet.

00:12 Let's determine some information about it.

00:14 So to determine any information, we need to model this mathematically.

00:18 We can write it as a geometric series.

00:21 That is, we have a1, the initial term, equal to 30 feet, and we have that r, the common ratio, is equal to 0 .8.

00:30 As each term is just multiplied by 0 .8 to get the next.

00:33 With this in mind, let's get started.

00:36 So how high is the third bounce going to be? let's find out using a formula from the book.

00:41 This formula states that the nth term, a .n, is going to be a1 times r to the n minus 1.

00:47 So if we let n equal 3, we get that a3 is equal to 30 times 0 .8 to the 30 minus 1, or to the, sorry, to the 3 minus 1, to the 2 power.

01:02 So now if you multiply this out, you will get that is equal to 19 .2 feet.

01:07 So on just the third bounce, we've already gone down over 10 feet to 19 .2 feet.

01:13 What's the generic form for just the nth bounce? well, we actually already found that.

01:19 Up here, it's the basic formula.

01:21 A -n equals a1 times r to the n -minus 1.

01:24 That wasn't so bad.

01:26 This next part gets pretty interesting.

01:28 So what is going to be the first bounce that is less than six inches? so six inches, we're dealing with feet.

01:36 So six inches, we can translate to 0 .5 feet.

01:41 And with this in mind, let's set up an equation.

01:44 So we don't know what n is going to be, but we know that a -n should be 0 .5.

01:50 And this is equal to 30 times r.

01:53 0 .8 to the n -1...

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