Problem 93 - Precalculus: Mathematics for Calculus

Question

A ball is dropped from a height of 9 ft. The elasticity of the ball is such that it always bounces up onethird the distance it has fallen. (a) Find the total distance the ball has traveled at the instant it hits the ground the fifth time. (b) Find a formula for the total distance the ball has traveled at the instant it hits the ground the \(n\) th time.

Step-by-Step Solution

Answer

(a) The ball travels approximately 17.963 ft by the 5th impact; (b) The total distance formula is \( D_n = 18 - 9(\frac{1}{3})^n \).

Step 1: Understanding the Sequence of Bounces

Every bounce after dropping the ball covers a distance equal to one-third of the previous fall distance. First, the ball falls 9 ft, and rises one-third of that height, which is 3 ft. Then, it again falls 3 ft and rises one-third of that new drop, which is 1 ft, and so on, with decreasing height of each subsequent bounce.

Step 2: Calculate the First Fall and Bounces

For the first fall, the ball drops 9 ft. On the first bounce, it rises to \(\frac{1}{3}\) of 9 ft, which is 3 ft, and falls back 3 ft. Hence, after the first bounce, the distance covered is \(9 + 3 + 3 = 15\) ft.

Step 3: Continue Calculating Subsequent Bounces

Continue the calculation for the second bounce where it rises \(\frac{1}{3}\) of 3 ft, that is 1 ft up and 1 ft down. So, the total distance traveled now is \(15 + 1 + 1 = 17\) ft. Follow the same logic for the third and fourth bounces, each time adding \(\frac{1}{3}\) of the last bounce to both the upward and downward paths.

Step 4: General Expression for Total Distance after nth Impact

To find a general formula, recognize the pattern: The ball covers the first drop plus two times the sum of the series of bounce heights. The total distance at the nth impact is given by \[ D_n = 9 + 2 \times \left(3 + \frac{1}{3} \times 3 + \frac{1}{3^2} \times 3 + \ldots + \frac{1}{3^{n-1}} \times 3 \right) \] Factoring out 3, \[ D_n = 9 + 6 \times \left(1 + \frac{1}{3} + \frac{1}{3^2} + \ldots + \frac{1}{3^{n-1}}\right) \] This series is a geometric series with a common ratio of \( \frac{1}{3} \). The sum of this series can be calculated as \[ S_n = \frac{1 - \left( \frac{1}{3} \right)^{n}}{1 - \frac{1}{3}} \] Therefore, the total distance is: \[ D_n = 9 + 6 \times \frac{1 - \left( \frac{1}{3} \right)^{n}}{\frac{2}{3}} \] Simplifying gives: \[ D_n = 9 + 9(1 - \left( \frac{1}{3} \right)^{n}) = 18 - 9 \left( \frac{1}{3} \right)^{n} \]

Step 5: Find the Total Distance at 5th Impact

Using the derived formula for total distance at the nth bounce, substitute \( n = 5 \) into \[ D_5 = 18 - 9 \left( \frac{1}{3} \right)^{5} \] Calculate the power term: \[ \left( \frac{1}{3} \right)^{5} = \frac{1}{243} \] Then, \[ D_5 = 18 - 9 \,\times\, \frac{1}{243} = 18 - \frac{9}{243} = 18 - \frac{1}{27} \] Approximately, \[ D_5 \approx 18 - 0.037 = 17.963 \]

Key Concepts

Elasticity in Physics

Elasticity refers to the ability of a material to return to its original shape or height after being deformed, such as when a ball bounces. In our exercise, the elasticity of the ball implies that it rebounds to one-third of the height from which it falls. This consistent behavior allows us to predict the motion of the bouncing ball.
Elastic objects like this ball follow basic principles of energy conservation. When the ball hits the ground, kinetic energy is converted into elastic potential energy, some of which allows the ball to bounce back. However, the fraction of the total mechanical energy lost per cycle determines how high the ball will bounce back, often expressed as a coefficient of restitution. In this case, the coefficient is \( \frac{1}{3} \) or one-third.
Understanding elasticity helps solve problems in physics where objects need to be modeled accurately in terms of their physical interactions and energy exchanges.

Sequences and Series

When dealing with problems involving repeated actions, such as the repetitive bounces of a ball, the concepts of sequences and series become very relevant. A sequence is an ordered list of numbers generated by some rule. Here, the bounces form a geometric sequence due to the constant ratio between consecutive terms.
The ball bounces back to \( \frac{1}{3} \) of its previous height. This forms a geometric series characterized by the base length decreasing consistently by the same ratio, \( \frac{1}{3} \). For example, after falling 9 ft initially, it bounces 3 ft, then 1 ft, and so on. To calculate the total distance traveled, you sum this sequence for both ascending and descending paths.

• The geometric series formula used here is \[ S_n = \frac{1 - r^n}{1 - r} \] where \( r \) is the common ratio.
• This series helps calculate total distances, essential in problems where motion or repetitive actions need evaluating.

Distance Traveled

To determine the distance the ball travels before stopping or reaching negligible bounce heights, one must consider both its upward and downward paths. Initially, it falls 9 ft and then follows upward and downward distances determined by a geometric sequence.
Calculating the total distance involves adding the initial drop to the sum of all subsequent upward and downward bounces. Each bounce height is determined as a part of the geometric sequence, repeatedly applying the elasticity ratio of \( \frac{1}{3} \). The formula derived gives us clarity:

• After the first drop, use the geometric series to find the total distance traveled until the nth hit: \[​D_n = 9 + 2 \sum \left( rac{1}{3}^{k-1} \cdot 3 \right) \]
• This simplification helps in solving for specific instances such as the 5th bounce back to the ground.

This application of mathematics offers a practical insight into predicting physical movements based on repeatedly occurring actions.

Problem-Solving in Mathematics

Mathematical problem-solving involves strategically tackling an issue to find a solution. It's about understanding concepts, identifying patterns, and applying mathematical principles to derive solutions, just like in the bouncing ball scenario.
The exercise demonstrates a structured approach, starting by understanding the problem and identifying the sequence of bounces. Calculating initial conditions, then generalizing to discover a formula using known mathematical concepts, such as geometric sequences, showcases critical problem-solving skills.

• Step-by-step calculation helps in systematically addressing each part of the problem, avoiding oversight.
• Finding patterns enables one to generalize results and offer solutions to similar problems efficiently.
• Applying proper formulae and recognizing relationships develops sound mathematical reasoning and critical thinking abilities.

This systematic approach to solving problems is vital in mathematics and applicable across various real-world scenarios, making it a key skill for students and professionals alike.