Problem 73 - Precalculus: Mathematical 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 1 \(\mathrm{m}\) above the ground, until it comes to rest.

Step-by-Step Solution

Answer

The ball travels a total of 3 meters.

Step 1: Understanding the problem

The ball is dropped from a height of 1 meter, and rebounds to half the height it previously reached each time. The total distance travelled includes the fall and the subsequent rise for each bounce, forming a geometric series.

Step 2: Define the geometric series components

The initial drop distance is 1 meter. Each complete bounce (fall and rise) after that forms a pair: the first bounce returns to 0.5 meters (upward and downward), the next goes up to 0.25 meters (upward and downward), and so on. This creates a series of distances.

Step 3: First drop and subsequent bounces

The first drop is 1 meter. Each bounce thereafter covers 2 times the height of the rebound (one upward, one downward). So, the first bounce contributes 2(0.5) meters, the second 2(0.25) meters, and continues as a geometric series.

Step 4: Forming the infinite series

The series can be written as: Total distance = 1 (initial drop) + 2(0.5) + 2(0.25) + 2(0.125) + ... This is an infinite geometric series where the first term after the initial drop is 2(0.5) and each subsequent term has a common ratio of 0.5.

Step 5: Apply the geometric series formula

The sum of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 2(0.5) = 1 \) and \( r = 0.5 \).

Step 6: Calculate the total distance

Using the formula, \( S = \frac{1}{1 - 0.5} = \frac{1}{0.5} = 2 \). However, this is the sum of all bounces excluding the initial drop. Adding the initial drop, the total distance is 1 (initial drop) + 2 (all bounces) = 3 meters.

Key Concepts

Infinite Series

An infinite series is essentially the summation of an infinite sequence of numbers. In this problem, the sum captures the total distance traveled by the ball after numerous bounces. The concept of adding an infinite number of terms might sound abstract, but it allows us to calculate definite values for every additional bounce, even as their individual heights decrease.
One important aspect of an infinite series is convergence, meaning that the series approaches a specific value, rather than diverging to infinity.
When dealing with distances as in our ball example, convergence assures us that the total distance traveled is finite, despite the countless bounces.

Geometric Progression

A geometric progression is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number, known as the common ratio.
In the case of the bouncing ball, the distances form a geometric progression. After the initial drop, each subsequent bounce reaches a height that is half of the previous bounce height, creating a common ratio of 0.5.
Here’s a simple breakdown:

• The ball first drops 1 meter. This is our starting point.
• It rebounds to 0.5 meters, and travels that distance both up and down.
• Next rebound is to 0.25 meters, again traveled twice (up and down).

This pattern continues, with each bounce's travel distance being 2 times the previous height reached.
Understanding geometric progression is crucial for forming the infinite geometric series utilized in calculating the total travel distance.

Calculus Applications

Calculus makes it possible to find sums of infinite series through formulated mathematical expressions. The sum of an infinite geometric series, like ours, can be succinctly found by using a formula.
For a geometric series with a first term, \( a \), and a common ratio, \( r \, -1 < r < 1\), the sum is given by \[ S = \frac{a}{1 - r} \]
In the ball example, the first rebound and subsequent bounces form the series. Here, \( a = 1 \) (from \( 2(0.5) \), the first bounce's double-path) and \( r = 0.5 \). Using this formula, the infinite series of bounces after the initial drop sums to 2 meters. Adding the initial drop (1 meter), you arrive at a total distance of 3 meters traveled.
Thus, calculus provides an elegant approach to quickly finding the total distance covered by accounting for every bounce in this infinite sequence.