Problem 92 - A Graphical Approach to Precalculus with Limits

Question

Height of a Dropped Ball Someone drops a ball from a height of 10 meters and notices that on each bounce the ball returns to about \(\frac{3}{4}\) of its previous height. About how far will the ball travel before it comes to rest? (Hint: Consider the sum of two sequences.) (IMAGE CANT COPY)

Step-by-Step Solution

Answer

The ball travels approximately 40 meters before coming to rest.

Step 1: Understanding the Problem

We need to find the total distance traveled by the ball. The ball's journey involves multiple bounces, each reaching a fraction of the previous height. The distance traveled can be represented as the sum of an infinite geometric series.

Step 2: Analyzing the Ball's Second Movement

After the ball is dropped from a height of 10 meters and bounces back to \( \frac{3}{4} \times 10 = 7.5 \) meters, it descends the 7.5 meters. We must also include this descent in our total. The descending distances form a geometric sequence.

Step 3: Adding Upwards and Downwards Distances

We decompose the total travel into the initial drop and the subsequent bounces (up and down). The sequence of movements after the first impact consists of a descending and ascending component, both multiplying by the common ratio each bounce.

Step 4: Recognizing the Geometric Series

The series for total distance after the initial drop is a geometric series: \[ 7.5 + 2(7.5 \rac{3}{4}) + 2(7.5 (\frac{3}{4})^2) + \ldots \] The series coefficient for the upward and downward distances is \frac{3}{4}, and it continues infinitely.

Step 5: Calculating Total Distance Traveled

The sum of the infinite series for the upward and downward components is given by \frac{1}{1-r}, where r is the common ratio of \frac{3}{4}. Calculate the sum as: \[ 7.5 (1 + 2(\frac{3}{4}) + 2(\frac{3}{4})^2 + \ldots) = 7.5 \rac{1}{1 - \rac{3}{4}} = 7.5 \ ime 4 = 30 \] Adding the original drop, the total path is \ 10 + 30 = 40 \ meters.

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In our ball bounce problem, the heights the ball reaches after each bounce form a geometric sequence.

• The initial height from which the ball is dropped is 10 meters.
• The common ratio in this scenario is \(\frac{3}{4}\), indicating that after each bounce, the ball reaches \(\frac{3}{4}\) of the height it achieved from the previous bounce.

The nature of the geometric sequence can be seen as:\[10, \ 10 \times \frac{3}{4}, \ 10 \times \left(\frac{3}{4}\right)^2, \ 10 \times \left(\frac{3}{4}\right)^3, \ldots\]Understanding this sequence helps in solving the problem, as it clearly represents the pattern of diminishing bounce heights.

Sum of Sequences

In mathematics, the sum of a sequence is the addition of all numbers in a sequence. With infinite geometric series, like our bounce problem, we can sum the sequence using a specific formula. For a geometric sequence with a first term \(a\) and common ratio \(r\), where \(|r| < 1\), the sum \(S\) of the infinite series is given by:\[S = \frac{a}{1 - r}\]In the bounce problem, once the ball is initially dropped, subsequent bounces up and down form an infinite geometric series:

• Each bounce's upward and downward path contribute to the sequence with the same common ratio and therefore add similarly to the sum.
• Using \(a = 7.5\) (the first bounce up or down) and \(r = \frac{3}{4}\), the sum for the upward and downward iterations becomes:
• \[ 2 \times 7.5 \left( \frac{1}{1-\left(\frac{3}{4}\right)} \right) = 30 \]

Add this calculated path to the initial 10 meters drop. Thus, the total travel distance is \(10 + 30 = 40\) meters.

Bounce Problem in Mathematics

The bounce problem is a classic example in mathematics, where a ball's movement is analyzed as it bounces up and down until it comes to rest. This illustrates the concept of a geometric sequence and its sum. The bounce problem commonly uses infinite geometric series since after each bounce, the heights decrease exponentially.

• The first drop is viewed separately as it is not part of the bounce sequence.
• The sequence only involves movements after each bounce.
• The mathematical model of the bounce sequences through geometric series allows for calculation of total distance without infinitely tracking each bounce.

By understanding this simplification, not only does one find the bounce problem intriguing, it becomes a demonstrative tool in teaching infinite sequences and series. The seamless reduction to a finite sum showcases the power of geometric reasoning in practical problem-solving, where infinitely repeating patterns converge to finite outcomes.