Problem 97 - Precalculus: Mathematics for Calculus

Question

A certain ball rebounds to half the height from which it is dropped. Use an infinite geometric series to approximate the total distance the ball travels after being dropped from I m above the ground until it comes to rest.

Step-by-Step Solution

Answer

The total distance traveled by the ball is 3 meters.

Step 1: Understanding the Problem

The ball is initially dropped from a height of 1 meter. It rebounds to half its previous height each time, which means this situation can be described using an infinite geometric series. We need to calculate the total distance the ball travels.

Step 2: Identify the Distance Segments

When the ball is dropped, it falls 1 meter, rebounds to half its height (0.5 meters), and falls again 0.5 meters. This process repeats infinitely. Thus, the segments of travel are: downward and upward movements.

Step 3: Calculate the First Few Distances

The first drop is 1 meter. The first rebound and drop is another 0.5 + 0.5 meters, then the ball rebounds 0.25 meters and drops another 0.25 meters, and so on.

Step 4: Identify the Series

The total distance can be expressed as: 1 (first fall) + 2(0.5) + 2(0.25) + 2(0.125) + ... The series for the rebounds is 0.5 + 0.25 + 0.125 + ...

Step 5: Define the Series

The series for the rebounds is a geometric series with the first term, \(a = 0.5\), and a common ratio \(r = 0.5\). The first term of the downfalls is not part of this geometric series.

Step 6: Use the Geometric Series Formula

The sum of an infinite geometric series \(a, ar, ar^2, ...\) is \(S = \frac{a}{1-r}\), where \(|r| < 1\). For the series, \( S = \frac{0.5}{1-0.5} = 1 \).

Step 7: Calculate Total Distance

The total distance is the first drop plus twice the sum of the series (since each rebound height is covered twice, once up and once down), so total distance \(= 1 + 2\times 1 = 3\) meters.

Key Concepts

geometric series formula

The geometric series formula is a crucial mathematical tool used to find the sum of a series where each term is a constant multiple of the previous one. This type of series is known as a geometric series. The formula for the sum of an infinite geometric series, when the absolute value of the common ratio \(|r|\) is less than one, is given by:\[S = \frac{a}{1-r}\]where:

• \(S\) is the sum of the infinite series.
• \(a\) is the first term of the series.
• \(r\) is the common ratio, or the factor that each term is multiplied by to produce the next term.

In the context of the exercise, the rebound heights of the ball form an infinite geometric series, with the first term being 0.5 (half the initial drop height) and the common ratio also being 0.5. This series represents the total up-and-down path traveled by the ball, excluding the initial drop. Understanding this formula and its application is key to efficiently solving problems related to infinite geometric series.

series convergence

Series convergence is a concept that helps us understand whether the terms of an infinite series add up to a finite number. For an infinite geometric series to converge, the absolute value of the common ratio \(|r|\) must be less than one. If the series converges, it means that as you sum more and more terms, the total sum approaches a specific number.

In our specific example of the bouncing ball, the common ratio is 0.5, which is clearly less than one, thus the series converges. This means that although theoretically, the ball will continue to bounce forever, the total distance it travels has a finite limit. Convergence is essential in determining the relevance and applicability of a series' sum to real-world problems, such as calculating rebound distance.

rebound distance calculation

Calculating the rebound distance of the ball involves using the principles of infinite geometric series and their convergence. Initially, the ball falls from a height of 1 meter. Upon hitting the ground, it rebounds to half of the previous height, starting a sequence of rebounding and falling that continues infinitely.

To find the total rebound distance after the first fall, we consider:

• The first fall of 1 meter. This distance is straightforward.
• Afterward, the ball rebounds to half the height, adding another sequence of drops and rebounds represented by an infinite geometric series: 0.5, 0.25, 0.125, and so on.
• The series for rebound can be calculated using the formula \( S = \frac{0.5}{1-0.5} = 1 \).

The total distance the ball travels is the sum of the initial drop and twice the sum of the rebound series (since each rebound height is traversed when going up and then coming back down). Therefore, the total distance is 1 meter + 2 meters from the infinite series, equaling 3 meters. This approach efficiently quantifies the complete motion path of the ball once rebounded.