Question

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it 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 \(r.\) a. Find the first four terms of the sequence of heights \(\left\\{h_{n}\right\\}.\) b. Find an explicit formula for the nth term of the sequence \(\left\\{h_{n}\right\\}.\) $$h_{0}=30, r=0.25$$

Short answer

Answer: The height of the ball after the fourth bounce is 0.1171875 meters.

Step-by-step solution

Define initial terms

Before we start calculating the heights after the first 4 bounces, let's define the initial values given in the exercise: Initial height, \(h_{0} = 30\)m, Fractional rebound height, \(r = 0.25\).

Calculate height after first bounce

To find the height after the first bounce, we simply multiply the initial height by the fractional rebound height: \(h_1 = h_0 * r\). Now substitute the given values: \(h_1 = 30 * 0.25 = 7.5\)m.

Calculate height after second bounce

Similarly, to find the height after the second bounce, multiply the height after the first bounce by the fractional rebound height: \(h_2 = h_1 * r\). Now substitute the known values: \(h_2 = 7.5 * 0.25 = 1.875\)m.

Calculate height after third and fourth bounce

Repeat the process to find the heights after the third and fourth bounces: \(h_3 = h_2 * r = 1.875 * 0.25 = 0.46875\)m. \(h_4 = h_3 * r = 0.46875 * 0.25 = 0.1171875\)m. So, the first four terms of the sequence are \(30, 7.5, 1.875, 0.46875, 0.1171875\).

Find an explicit formula for the nth term

To find a general formula for the nth term of the sequence, notice that each term is obtained by multiplying the previous term by the fractional rebound height, \(r\). Therefore, the nth term can be calculated as: \(h_n = h_0 * r^n\). So, the explicit formula for the nth term of the sequence is \(h_n = 30 * 0.25^n\).

Key concepts

Series and Sequences

Understanding series and sequences is essential to grasp how values change over time. In mathematics, a sequence is an ordered list of numbers following a specific rule. Each number in a sequence is referred to as a term. When these sequences sum up specific terms, they form a series. There are various types of sequences, with geometric sequences being a common example.

A geometric sequence is one in which each term is found by multiplying the previous term by a constant factor. This factor is known as the common ratio, denoted as \(r\). If you have a starting number, called the initial term, you can find subsequent terms by repeatedly multiplying by the common ratio. In our example, the sequence of heights that a ball reaches after each bounce forms a geometric sequence. This is because each new height is a consistent fraction of the height before it.

In general, the formula for any term in a geometric sequence is given by \(h_n = h_0 \times r^n\), where \(h_0\) is the initial term and \(r^n\) represents the common ratio raised to the nth power. Understanding this concept allows you to predict the behavior of the sequence accurately.

Rebound Height

In the context of a bouncing ball, the rebound height refers to how high the ball bounces back after it hits the ground. If you throw a ball upwards, it will fall and bounce back up. However, unless it hits a super bouncy trampoline, the ball won't reach its original height.

The rebound height is typically a fraction of the height from which the ball fell. This fraction is determined by the properties of the ball and the surface it bounces on. For instance, if a ball is dropped from a height of 30 meters and rebounds to 7.5 meters after the first bounce, the fractional rebound is 0.25, or 25% of its previous height. This calculation is achieved by multiplying the previous height by the fractional rebound height \(r\).

In the example we have, the sequence of rebound heights starts with 30 meters, and with each bounce, it decreases consecutively to 7.5 meters, 1.875 meters, and so on. Understanding rebound height helps in grasping why the ball slows down as it continues to bounce.

Fractional Rebound

The fractional rebound is an important factor in determining how high a ball will bounce back after hitting a surface. It's defined as the ratio between the height a ball reaches after bouncing, and the height from which it initially fell. In mathematical terms, the fractional rebound can be represented by the variable \(r\) in a geometric sequence.

In simpler terms, if you have a fraction like 0.25 for \(r\), it means that the ball rises to 25% of the height from its previous bounce. This percentage remains constant for every bounce if the conditions do not change. This constancy makes it easy to calculate subsequent heights after each bounce.

• First Bounce: Multiply the initial height by the fractional rebound.
• Second Bounce: Multiply the outcome of the first bounce by the fractional rebound again.
• Repeat the process for subsequent bounces.

This consistency is crucial when trying to find how many bounces it would take for the ball to come to a near stop or when estimating how much time it would take to do so.