Problem 136 - Precalculus with Limits

Question

A ball is dropped from a height of \( 16 \) feet. Each time it drops \( h \) feet, it rebounds \( 0.81 h \) feet. (a) Find the total vertical distance traveled by the ball. (b) The ball takes the following times (in seconds) for each fall. \( s_1 = -16t^2 + 16,\quad \quad \quad \quad \quad s_1 = 0 if t = 1 \) \( s_2 = -16t^2 + 16\left(0.81\right), \quad \quad s_2 = 0 if t = 0.9 \) \( s_3 = -16t^2 + 16\left(0.81\right)^2, \quad \quad s_3 = 0 if t = \left(0.9\right)^2 \) \( s_4 = -16t^2 + 16\left(0.81\right)^3, \quad \quad s_4 = 0 if t = \left(0.9\right)^3 \) \( \vdots \quad \quad \quad \quad \quad \quad \quad\vdots \) \( s_n = -16t^2 + 16\left(0.81\right)^{n - 1}, \quad s_n = 0 if t = \left(0.9\right)^{n-1} \) Beginning with \( s_2 \), the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is \( t = 1 + 2 \sum_{n=1}^{\infty}\left(0.9\right)^n \) Find this total time.

Step-by-Step Solution

Answer

Firstly, substitute values into the formula for the sum of the infinite series to get the total vertical distance. Secondly, compute the total bounce time by substituting values into the formula for the sum of the infinite series.

Step 1: Compute the total distance travelled by the ball

We can sum up the distances travelled by the ball on each descend and rebound. The initial height from where the ball is dropped is 16 feet. Each rebound reaches \(0.81\) as high as the previous height. So, an ascending series describes these distances travelled, which is \(16+2(0.81\times16)+2\left(0.81\right)^2\times16+\cdots\). This series has first term \(a = 16\), common ratio \(r = 0.81\times2\) and is infinite.

Step 2: Calculate Total Distance

The sum of an infinite geometric series can be obtained using the formula \(S=a/(1-r)\). Here \(a = 16\) and \(r = 0.81\times2\). Substituting these values into the formula, we get \(S = 16/(1 - 0.81\times2)\). After calculating this, we get the total vertical distance travelled by the ball.

Step 3: Compute the total bounce time

We divide the motion of the ball into two: downward and upward bounce, each taking the same time after the first bounce. Each bounce time sequence forms a geometric series \(1 + 2 \sum_{n=1}^{\infty}\left(0.9\right)^n \). It is an infinite series with first term \(a = 0.9\), common ratio \(r = 0.9\), and an additional first downfall time 1.

Step 4: Calculate Total Bounce Time

The sum of this infinite series can be computed using the formula \(S = t + 2\times(a/(1-r))\), where the first falling time \(t = 1\), \(a = 0.9\), and \(r = 0.9\). Substituting these values into the formula gets the total bounce time.

Key Concepts

Infinite Geometric Series

When a bouncing ball hits the ground and rebounds to a fraction of its previous height repeatedly, we can represent the total distance travelled as an infinite geometric series. In such a series, each term after the first is found by multiplying the previous term by a constant called the common ratio. For the ball dropped from a height of 16 feet and rebounding to 0.81 times its previous height, the common ratio is 1.62 (since the ball travels up and down over the same distance).
To find the total distance, we use the formula for the sum of an infinite geometric series, which is \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio, provided \( \left| r \right| < 1 \). Otherwise, the series does not converge, meaning it doesn't add up to a finite number. Here, with \( a = 16 \) and \( r = 1.62 \times 0.81 \), the value of \( r \) is less than 1, indicating the series converges and we can calculate the total distance using the formula.
Understanding infinite geometric series is vital for solving problems involving repeated actions that diminish or grow by a constant proportion, such as depreciation, interest rates, or, like in our case, bouncing objects.

Series Convergence

For a series to be useful in calculations, especially in physics or financial models, it must converge, which means the sequence of partial sums of the series approaches a specific value as more and more terms are added. The infinite geometric series converges if the absolute value of the common ratio \( r \) is less than one; otherwise, it's called a divergent series and the sum is infinite. In the bouncing ball exercise, the series converges because the common ratio is the product of \( 0.81 \), due to the rebound, and \( 2 \) because the distance is covered twice every bounce after the initial drop, resulting in a value less than one.

Importance of Convergence

Convergence is a core concept for many mathematical and real-world applications. It ensures that we can make sense of the behavior of infinite processes by analyzing their finite sums. Whether we are looking at the total distance traveled by a ball or projecting the future growth of an investment, it allows us to work with infinite sums in a meaningful way.

Quadratic Equations

Quadratic equations take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. The ball’s motion as it falls and rebounds is modeled by a quadratic equation in the given problem, with the height \( s \) as a function of time \( t \). The times at which the ball hits the ground, leading to \( s = 0 \), correspond to the solutions of these quadratic equations. For example, the equation \( -16t^2 + 16 = 0 \) for the first fall can be factored to find the time of impact, which is at \( t = 1 \) second.

Applications of Quadratic Equations

Quadratic equations are widely used to model phenomena that involve a squared relationship between two quantities, such as area calculations, projectile motions, and optimizing economic functions. In physics, they are especially useful in describing parabolic trajectories, like the path of our bouncing ball.