Question

A ball is thrown upward to a height of \(h_{0}\) meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce. Consider the following values of \(h_{0}\) and \(r\). $$h_{0}=30, r=0.25$$

Short answer

Answer: To find the height of the ball after the third bounce, we can plug n=3 into the given formula: $$h_{3} = 30 \cdot 0.25^3$$ $$h_{3} = 30 \cdot 0.015625$$ $$h_{3} = 0.46875 \,\text{m}$$ The height of the ball after the third bounce is approximately 0.47 meters.

Step-by-step solution

Identify the given values

First, we identify the given values for the problem: the initial height, \(h_{0} = 30 \,\text{m}\), and the rebound fraction, \(r = 0.25\).

Understand the rebound relationship

Every time the ball bounces, it reaches a certain percentage of the previous height. From the given information, we know that after each bounce, the ball reaches 25% (0.25) of the previous height it was at.

Formulate the height relationship

Since after each bounce, the height of the ball is a fraction \(r\) of the previous height, we can represent the height after the nth bounce as a geometric progression with the form: $$h_{n} = h_{0} \cdot r^n$$ Where \(h_{n}\) is the height after the nth bounce, \(h_{0}\) is the initial height, and \(r\) is the rebound fraction.

Find the height after the first bounce

To find the height of the ball after the first bounce, we can plug the values for \(h_{0}\) and \(r\) into the formula: $$h_{1} = 30 \cdot 0.25^1$$ $$h_{1} = 30 \cdot 0.25$$ $$h_{1} = 7.5 \,\text{m}$$ So, the height of the ball after the first bounce is 7.5 meters.

Find the height after any nth bounce

Since we have the formula for the height after the nth bounce, we can easily calculate the height at any given bounce: $$h_{n} = h_{0} \cdot r^n$$ In this case, $$h_{n} = 30 \cdot 0.25^n$$

Key concepts

Understanding Rebound Fraction

Whenever an object, like a ball, bounces off a surface, it loses energy and bounces back to a height that is a certain fraction of the height from which it fell. This consistency leads us to the rebound fraction (r), a crucial factor in calculating successive heights in physics and mathematics.

The rebound fraction is a ratio between 0 and 1 that represents the proportion of height a ball reaches after each bounce compared to the previous bounce. For example, if a ball bounces back to a quarter of its previous height, the rebound fraction is given as 0.25. This simple, yet important concept helps us understand the diminishing heights of the bounces, which can be formulated into a geometric sequence.

In the given exercise, every bounce the ball makes is to a height that is only 25% of the height of the preceding bounce. This is crucial in predicting the ball's behavior over multiple bounces and is the first step in recognizing the pattern formed by the heights of the bounces.

Diving Into Geometric Sequences

A geometric sequence is a series 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. Geometric sequences can describe many phenomena in real life, such as the rebound heights of a ball.

The formula for the nth term of a geometric sequence is given by:
\[a_n = a_1 \times r^{(n-1)}\]
where \(a_n\) represents the nth term, \(a_1\) is the first term, and \(r\) is the common ratio.

In our exercise, the initial height, \(h_0\), acts as the first term of the sequence, and the rebound fraction \(r\) is the common ratio. Using these, we can model the ball's behavior as a geometric sequence to find the height after any number of bounces. This powerful concept demonstrates predictability in what may otherwise seem like a complex physical process.

Exploring Exponential Decay in Bouncing Balls

The concept of exponential decay is widely found in physics, chemistry, biology, and finance, wherever a quantity decreases at a rate proportional to its current value. In the context of the bouncing ball, exponential decay perfectly illustrates how the ball's height decreases over successive bounces.

The mathematical representation of exponential decay is similar to that of a geometric sequence and is given by:
\[N(t) = N_0 \times e^{-kt}\]
where \(N(t)\) is the remaining quantity at time \(t\), \(N_0\) is the initial quantity, \(e\) is the base of the natural logarithm, and \(k\) is the decay constant. In the case of our exercise, the decay is represented through the formula \(h_{n} = h_{0} \times r^n\), with each bounce of the ball being analogous to a 'time step' in exponential decay models.

Through this lens, we can appreciate the mathematical beauty behind a simple action of a ball bouncing and grasp the broader implications of exponential decay, a concept that we find from radioactivity to depreciation in asset values.