Problem 73 - Algebra and Trigonometry

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 \(n\) th bounce.

Step-by-Step Solution

Answer

The ball rebounds to approximately 18.98 ft on the fifth bounce. The formula for the nth bounce height is \( h_n = 80 \times (0.75)^n \).

Step 1: Determine Initial Conditions

The ball is initially dropped from a height of 80 ft. This height is used to calculate the rebound height after the first bounce, which will be used as the pattern continues.

Step 2: Understand the Formula for Rebound Height

Each time the ball bounces, it rebounds to three-fourths of the previous height. Thus, the rebound height can be calculated using the formula: \[ h_n = h_0 \times (r)^n \]where \( h_0 \) is the initial height (80 ft), \( r \) is the rebound ratio (0.75), and \( n \) is the bounce number.

Step 3: Apply Formula to Calculate Fifth Bounce

Substitute the values into the rebound formula to find the height after the fifth bounce.Using \( h_0 = 80 \), \( r = 0.75 \), and \( n = 5 \), the formula becomes:\[ h_5 = 80 \times (0.75)^5 \]

Step 4: Calculate Numerical Solution

Compute the expression:\[ h_5 = 80 \times (0.75)^5 = 80 \times 0.2373046875 = 18.984375 \]Thus, the height the ball rebounds on the fifth bounce is approximately 18.98 ft.

Step 5: Generalize Formula for nth Bounce

The general formula for the rebound height on the nth bounce is:\[ h_n = 80 \times (0.75)^n \]This represents the exponential decay of the bounce heights at each step.

Key Concepts

Rebound Height Formula

When a ball is dropped, understanding how high it will bounce back is all about the Rebound Height Formula. This formula helps us figure out the height of the ball after each bounce. Let's break it down:

- **Initial Height (\( h_0 \))**: This is the starting height from which the ball is dropped. In our example, it's 80 feet.
- **Rebound Ratio (\( r \))**: This represents the fraction of the drop height that the ball recovers with each bounce. Here, it's 0.75, meaning the ball rebounds to 75% of its previous height.
- **Bounce Number (\( n \))**: This is just the number of bounces. We calculate the height after each bounce with this number.

The rebound height for any bounce, \( n \), is found by the formula:\[ h_n = h_0 \times (r)^n \]This equation shows how with each bounce, the height reduces exponentially. By plugging the specifics of our problem into the formula, namely \( h_0 = 80 \) and \( r = 0.75 \), you can calculate how high the ball will rebound after any given bounce.

Elasticity in Physics

Elasticity isn't just about rubber bands and stretching materials — in physics, it relates to how energy transfers when objects collide. For the ball bouncing scenario, elasticity dictates how well the ball can return to its pre-collision state, or height in this situation.

In our bouncing ball case, the elasticity of the ball is quantified by the rebound ratio (0.75). This ratio indicates that the ball retains 75% of its vertical energy with each bounce. The other 25% of the energy might be lost to factors like air resistance or heat due to friction.

Here's why elasticity is crucial:

• It leads to **predictable patterns** in the behavior of bouncing objects allowing us to use formulas like the rebound height formula to predict outcomes.
• It's important in **designing materials** and products, such as balls, car bumpers, and more, where controlled energy absorption is crucial.

Without understanding elasticity, predicting how high the ball would rebound would be more guesswork than science!

Geometric Sequence

Any pattern where each term is derived from the previous one by multiplying by a constant is known as a Geometric Sequence. The concept is central to our bouncing ball exercise.

Here's how it connects:

• **Initial Term (\( a \))**: The first term in the bouncing series is the initial height of the bounce, which is 80 feet.
• **Common Ratio (\( r \))**: This is the factor we multiply by to get the next term in the sequence, which in this case, is the rebound ratio, 0.75.
• Each bounce creates a new term by applying the common ratio to the previous term, hence forming a geometric sequence.

In a Geometric Sequence, each term is found by multiplying the previous term by the common ratio \((r)\). The formula for any term \( n \) in such a sequence is:\[ a_n = a \times r^{(n-1)} \]
This sequence helps us predict how high the ball bounces at each step, clearly illustrating the exponential decay in height with successive bounces. Recognizing this sequence allows us to understand the systematic decline in the ball's bounce height efficiently.