Question

A rubber ball dropped from a height of exactly \(6 \mathrm{ft}\) bounces (hits the floor) several times, losing \(10 \%\) of its kinetic energy each bounce. After how many bounces will the ball subsequently not rise above \(3 \mathrm{ft}\) ?

Short answer

The ball will not rise above 3 feet after 7 bounces.

Step-by-step solution

Identify the pattern

After every bounce, the ball retains only 90% of its previous height. Hence, its height drops to 90% of the previous height every time, i.e., \(height = 0.9 \times previous\ height\)

Establish the iterative calculation

Start with an initial height of 6 feet and perform the calculation repeatedly. After each bounce, calculate the new height by taking 90% (0.9) of the current height.

Stop when the height is less than 3 feet

Continue the iterations until the result is less than 3 feet. The number of iterations is equal to how many bounces occurred before the ball height dropped below 3 feet.

Key concepts

Conservation of Energy

The law of conservation of energy is a fundamental principle that states energy cannot be created or destroyed, only transformed from one form to another. In the context of a bouncing ball, as the ball falls, its potential energy (due to its height) is converted into kinetic energy (energy of motion). When the ball hits the ground, some of this kinetic energy is transformed back into potential energy as the ball rises, but not all. Some energy is always lost, often as heat due to friction and deformation of the ball and surface.

When discussing bounces, it's essential to understand that no bounce is perfectly efficient; energy losses are inevitable. These losses manifest as lower heights achieved by the ball after each bounce. Recognizing how kinetic energy diminishes with each bounce helps us predict the behavior of the ball and calculate parameters such as the height of subsequent bounces.

In an exercise, like understanding the height a ball will achieve after several bounces, conservation of energy serves as the underlying concept that guides the formation of mathematical models and equations used to solve the problem.

Elastic Collisions

An elastic collision is an event where, after collision, the total kinetic energy is the same as it was before the event. In these collisions, objects bounce back without suffering loss in the total kinetic energy. However, in real-life scenarios, perfectly elastic collisions don't exist; some energy is always transformed into other forms, like sound or heat.

This is pertinent to our rubber ball, which undergoes what's called an inelastic collision when it bounces. Each collision with the ground leads to a kinetic energy loss, reflected by the ball rising to a lower height after each bounce. This concept is essential when estimating how a ball's bounce height decreases over time. It helps predict the outcome of repetitive impacts—such as balls bouncing or cars in bumper-to-bumper traffic—not just looking at the first impact but considering a series of impacts.

Percent Decrease

Percent decrease is a mathematical concept used to quantify the reduction in a value by a specific percentage. It's calculated by comparing the difference between the original value and the reduced value to the original value.

For example, if a ball loses 10% of its kinetic energy with each bounce, this means that if it had 100 units of energy to start with, it would have 90 units after the first bounce, 81 units after the second (which is 90% of 90), and so on. The percent decrease provides a consistent way to represent the energy loss and can be applied successively to calculate the progressive reduction in height after each bounce. While simple when calculating one or two bounces, employing the concept to a series of bounces illustrates the compounded effect of energy loss over time and is crucial in solving problems that involve iterative energy loss or gains.

Iterative Calculations

Iterative calculations are repetitive computational processes used to approach a desired outcome or solution. Each step in the iteration is based on the previous one, making it particularly useful for problems with ongoing processes like a ball bouncing and losing energy each time.

In the given exercise, the iterative calculation involves multiplying the ball's height by 0.9 (or subtracting a 10% loss) after each bounce. With each iteration, the height reduces until it falls below the defined threshold of 3 feet. Iterative processes not only provide the means to determine outcomes over multiple cycles but also illustrate how gradual changes accumulate over time—essential for understanding phenomena in physics, economics, biology, and many other fields.