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Chapter 8: Problem 68

A ball is thrown upward to a height of \(h_{0}\) meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine \(a\) plausible value for the limit of \(\left\\{S_{n}\right\\}\) $$h_{0}=20, r=0.5$$

Short Answer

Expert verified
Answer: A plausible value for the limit of the sequence \(\left\\{S_{n}\right\\}\) in this case is 60 meters.

Step by step solution

01

Calculate the first four heights after each bounce

Using the initial height (\(h_{0}=20\)) and the given fraction (\(r=0.5\)), we can calculate the height after the first bounce. Since the ball rebounds to a fraction of its previous height, we can find the height by multiplying the fraction by the initial height: \(h_{1}=h_{0}r=20(0.5)=10\) To find the heights after the other bounces, we multiply the height of the previous bounce by the same fraction, \(r\). \(h_{2}=h_{1}r=10(0.5)=5\) \(h_{3}=h_{2}r=5(0.5)=2.5\) \(h_{4}=h_{3}r=2.5(0.5)=1.25\)
02

Determine the total distance the ball has traveled for each bounce

To find \(\left\\{S_{n}\right\\}\), we will add up the distance the ball has traveled during each bounce. The total distance consists of the distance the ball travels when rising in height and falling to the ground. We can calculate this by adding the heights, \(h_0\), \(h_1\), \(h_2\), \(h_3\), and so on. $$ S_{1}=h_{0}+h_{1}=20+10=30 $$ $$ S_{2}=S_{1}+2h_{2}=30+2(5)=40 $$ $$ S_{3}=S_{2}+2h_{3}=40+2(2.5)=45 $$ $$ S_{4}=S_{3}+2h_{4}=45+2(1.25)=47.5 $$ So the first four terms of \(\left\\{S_{n}\right\\}\) are; \(30\), \(40\), \(45\), and \(47.5\) meters.
03

Generate a table of terms for \(\left\\{S_{n}\right\\}\) and determine the plausible limit

We can continue to calculate the total distance traveled after more bounces using the geometric decay of \(h_n\). After calculating twenty terms, you'll notice that the total distance traveled is converging towards a value. In this case, that value is 60 meters. Therefore, a plausible value for the limit of the sequence \(\left\\{S_{n}\right\\}\) is 60 meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence
In mathematics, a sequence is an ordered list of numbers. When dealing with geometric sequences, each term is a product of the previous term and a constant ratio. In this context, the ball's path after each bounce creates a geometric sequence.

With the initial height of the ball (\(h_0=20\) meters) and the rebound fraction (\(r=0.5\)), every subsequent height \(h_n\) can be calculated using the formula: \[ h_n = h_{n-1} imes r \]

This means: - First bounce: \( h_1 = 20 imes 0.5 = 10 \) meters
- Second bounce: \( h_2 = 10 imes 0.5 = 5 \) meters
- Third bounce: \( h_3 = 5 imes 0.5 = 2.5 \) meters

This process of multiplying by a common ratio is what defines the sequence as a geometric one. As you can see, the sequence decreases over time, leading us to the next important concept in sequences: their limit.
Limit
The concept of a limit is a fundamental building block in calculus and analysis. A limit describes the value that a sequence approaches as the term number becomes very large. In the context of our problem, it is crucial to find the limit of the geometric sequence representing the ball's total distance traveled over time.

As you calculate more terms of the sequence \(\{S_n\}\), each representing the total distance traveled after each bounce, you will notice a pattern; the values are getting closer to 60 meters. This suggests that the sequence is converging to this number.

Formally, the limit of the infinite series for the total distance (when \(n\) goes to infinity) can be expressed as:
\[ S = \frac{h_0}{1 - r} \]

Plug in 20 for \(h_0\) and 0.5 for \(r\), we see:
\[ S = \frac{20}{1 - 0.5} = 60 \] meters

This result confirms the observation that the total distance approaches 60 meters. Thus, we interpret this as the theoretical maximum total path covered by the ball, with each bounce adding a smaller increment of travel.
Total Distance Traveled
When we discuss the total distance traveled in this bouncing ball problem, we refer to both the upward and downward journeys after each bounce.

Initially, the ball is thrown up, and it falls back down covering \(h_0\) height. After its first bounce, it reaches \(h_1\) and then returns to the ground, adding two times \(h_1\) to the total distance. Taking all bounces into account, the calculation for total distance becomes:
  • \( S_1 = h_0 + h_1 \)
  • \( S_2 = S_1 + 2h_2 \)
  • \( S_3 = S_2 + 2h_3 \)
and so on.
As we add these values, we observe the series grows by smaller amounts as \(h_n\) becomes smaller due to the factor \(0.5\).

The total distance can be pictured as the summation of an infinite geometric series which tends towards a finite sum, known as the limit, as discussed above.
By understanding these calculations, one sees how the series mirrors the physical process of a ball bouncing and gradually coming to rest after traversing a total distance.

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Most popular questions from this chapter

An insulated window consists of two parallel panes of glass with a small spacing between them. Suppose that each pane reflects a fraction \(p\) of the incoming light and transmits the remaining light. Considering all reflections of light between the panes, what fraction of the incoming light is ultimately transmitted by the window? Assume the amount of incoming light is 1.

The prime numbers are those positive integers that are divisible by only 1 and themselves (for example, 2,3,5,7, 11,13, \(\ldots\) ). A celebrated theorem states that the sequence of prime numbers \(\left\\{p_{k}\right\\}\) satisfies \(\lim _{k \rightarrow \infty} p_{k} /(k \ln k)=1 .\) Show that \(\sum_{k=2}^{\infty} \frac{1}{k \ln k}\) diverges, which implies that the series \(\sum_{k=1}^{\infty} \frac{1}{p_{k}}\) diverges.

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{75^{n-1}}{99^{n}}+\frac{5^{n} \sin n}{8^{n}}$$

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\tan ^{-1}\left(\frac{10 n}{10 n+4}\right)$$

a. Sketch the function \(f(x)=1 / x\) on the interval \([1, n+1]\) where \(n\) is a positive integer. Use this graph to verify that $$\ln (n+1)<1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}<1+\ln n.$$ b. Let \(S_{n}\) be the sum of the first \(n\) terms of the harmonic series, so part (a) says \(\ln (n+1)0,\) for \(n=1,2,3, \ldots\) c. Using a figure similar to that used in part (a), show that $$\frac{1}{n+1}>\ln (n+2)-\ln (n+1).$$ d. Use parts (a) and (c) to show that \(\left\\{E_{n}\right\\}\) is an increasing sequence \(\left(E_{n+1}>E_{n}\right)\). e. Use part (a) to show that \(\left\\{E_{n}\right\\}\) is bounded above by 1 . f. Conclude from parts (d) and (e) that \(\left\\{E_{n}\right\\}\) has a limit less than or equal to \(1 .\) This limit is known as Euler's constant and is denoted \(\gamma\) (the Greek lowercase gamma). g. By computing terms of \(\left\\{E_{n}\right\\}\), estimate the value of \(\gamma\) and compare it to the value \(\gamma \approx 0.5772 .\) (It has been conjectured that \(\gamma\) is irrational.) h. The preceding arguments show that the sum of the first \(n\) terms of the harmonic series satisfy \(S_{n} \approx 0.5772+\ln (n+1)\) How many terms must be summed for the sum to exceed \(10 ?\)
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