Problem 65
Question
Bouncing Ball A ball is dropped from a height of 80 \(\mathrm{ft} .\) The elasticity of this ball is such that it rebounds three-fourths of the distance it has fallen. How high does the ball rebound on the fifth bounce? Find a formula for how high the ball rebounds on the \(n\) the bounce.
Step-by-Step Solution
Verified Answer
The ball rebounds approximately 18.98 feet on the fifth bounce. The formula is \( h_n = 80 \times \left(\frac{3}{4}\right)^n \).
1Step 1: Understanding the Problem
The ball is initially dropped from a height of 80 feet. Each time it hits the ground, it rebounds to three-fourths, or 75%, of the height it fell from. We need to determine the height after the fifth bounce and find a general formula for the nth bounce.
2Step 2: Setting Up the Formula
To find the height after each bounce, we use the formula for geometric sequences. The initial height is 80 feet, and each subsequent height is 75% of the previous height. Thus, the height after the nth bounce can be described by the equation: \[ h_n = 80 \times \left(\frac{3}{4}\right)^n \] where \( h_n \) is the height after the nth bounce.
3Step 3: Calculating the Fifth Bounce
To find the height after the fifth bounce, we substitute \( n = 5 \) into our formula: \[ h_5 = 80 \times \left(\frac{3}{4}\right)^5 \] This means calculating \( 80 \times (0.75)^5 \).
4Step 4: Performing the Calculation
Calculate \( (0.75)^5 \), which equals approximately 0.2373. Then multiply this by 80: \[ h_5 = 80 \times 0.2373 = 18.984 \text{ feet} \] Thus, on the fifth bounce, the ball rebounds to approximately 18.98 feet.
5Step 5: General Formula Conclusion
From our steps, we derived the nth bounce formula: \[ h_n = 80 \times \left(\frac{3}{4}\right)^n \] This formula allows us to compute the rebound height for any bounce number n.
Key Concepts
Bouncing Ball ProblemElasticity and Rebound HeightMathematical Modeling
Bouncing Ball Problem
The bouncing ball problem introduces an exciting scenario where we observe the behavior of a ball after being dropped from a certain height. As the ball hits the ground, it rebounds to a fraction of the original height due to the ball's elasticity. This kind of problem is not only a fun illustration of physics in action but also provides a practical example of geometric sequences in mathematics.
In this specific case, the ball is initially dropped from a height of 80 feet. Elasticity determines how much of the initial height the ball retains in its rebound. Understanding real-world applications, like bouncing balls, helps build intuition about how mathematical concepts can describe natural phenomena through predictable patterns.
In this specific case, the ball is initially dropped from a height of 80 feet. Elasticity determines how much of the initial height the ball retains in its rebound. Understanding real-world applications, like bouncing balls, helps build intuition about how mathematical concepts can describe natural phenomena through predictable patterns.
Elasticity and Rebound Height
Elasticity is a measure of how well a material returns to its original shape after deformation. In the context of our bouncing ball problem, elasticity reflects the ball's capability to rebound after hitting the ground.
Each time the ball hits the ground, it rebounds three-fourths of the height from which it fell. This fractional rebound height is integral to solving the problem because it establishes the consistent reduction in height with each bounce. This reduction factor can be seen mathematically as the common ratio in a geometric sequence. In many homework problems involving balls and elasticity, you will often compute successive rebound heights using this concept of elasticity.
Each time the ball hits the ground, it rebounds three-fourths of the height from which it fell. This fractional rebound height is integral to solving the problem because it establishes the consistent reduction in height with each bounce. This reduction factor can be seen mathematically as the common ratio in a geometric sequence. In many homework problems involving balls and elasticity, you will often compute successive rebound heights using this concept of elasticity.
- Initial height: 80 feet
- Rebound fraction: 0.75 of previous height
Mathematical Modeling
Mathematical modeling is crucial for understanding complex systems by simplifying them into mathematical terms. In the case of our bouncing ball exercise, a geometric sequence model is employed to compute the subsequent heights of the ball after each bounce.
The general formula derived, \( h_n = 80 \times \left(\frac{3}{4}\right)^n \), beautifully illustrates how mathematics provides a tool for prediction. Here, \( h_n \) represents the height after the \(n\)th bounce, where 80 is the initial height, and \(\frac{3}{4}\) reflects the elasticity percentage of each rebound compared to the previous height.
This model not only aids in solving the current problem but also equips students to tackle similar questions involving different initial conditions or elasticity factors. By understanding how to build and manipulate such a model, students can predict outcomes and solve various iterative problems.
The general formula derived, \( h_n = 80 \times \left(\frac{3}{4}\right)^n \), beautifully illustrates how mathematics provides a tool for prediction. Here, \( h_n \) represents the height after the \(n\)th bounce, where 80 is the initial height, and \(\frac{3}{4}\) reflects the elasticity percentage of each rebound compared to the previous height.
This model not only aids in solving the current problem but also equips students to tackle similar questions involving different initial conditions or elasticity factors. By understanding how to build and manipulate such a model, students can predict outcomes and solve various iterative problems.
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