Problem 65 - Precalculus: Mathematical for Calculus

Question

A ball is dropped from a height of 80 ft. The elasticity of this ball is such that it rebounds threefourths 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 18.98 ft on the fifth bounce. The formula for the nth bounce is \( h_n = 80 \times \left( \frac{3}{4} \right)^n \).

Step 1: Understanding the Problem

We have a ball that is dropped from a height of 80 ft. Upon bouncing, it rebounds to three-fourths of the height from which it fell. We need to calculate the height of the fifth rebound and find a general formula for any nth bounce.

Step 2: Calculating the Rebound Heights

To find how high the ball rebounds, we multiply the previous height by the rebound ratio of \( \frac{3}{4} \). Initially, the height is 80 ft, and the first rebound is \( 80 \times \frac{3}{4} = 60 \) ft.

Step 3: Calculating the Second Rebound

Using the first rebound height of 60 ft, we calculate the second rebound height: \( 60 \times \frac{3}{4} = 45 \) ft.

Step 4: Calculating Subsequent Rebounds

Continuing with this pattern, the third rebound height is \( 45 \times \frac{3}{4} = 33.75 \) ft, and the fourth is \( 33.75 \times \frac{3}{4} = 25.3125 \) ft.

Step 5: Finding the Fifth Rebound

For the fifth rebound, multiply the height of the fourth rebound by \( \frac{3}{4} \): \( 25.3125 \times \frac{3}{4} = 18.984375 \) ft.

Step 6: Deriving the General Formula

Observe that each bounce is computed as a geometric sequence. Hence, the height after the nth bounce can be represented as: \[ h_n = 80 \times \left( \frac{3}{4} \right)^n \], where 80 ft is the initial height and \( \frac{3}{4} \) is the rebound ratio.

Key Concepts

Rebound Height

When we talk about the rebound height of a ball, we're essentially discussing how high a ball bounces back after hitting the ground. Initially, the ball falls from a certain height, and due to its elasticity, it rebounds, but not to the original height.

• The rebound height depends on the property of the ball, namely its elasticity.
• If, for example, a ball rebounds to three-fourths of its pre-drop height, each bounce will be only 75% of the previous height.
• This means that after every bounce, the ball reaches a progressively lower height.

Understanding the concept of rebound height helps in predicting the behavior of the ball over multiple bounces. In the given exercise, a ball starts at a height of 80 ft and rebounds to 75% of the height from which it has fallen each time. As you calculate subsequent bounces, you'll see a clear pattern emerge.

Geometric Progression

Geometric progression is a mathematical concept where each term after the first is derived by multiplying the previous one by a constant called the common ratio. For our bouncing ball exercise, the sequence of rebound heights forms a geometric progression.

• Starting with an initial height of 80 ft, each subsequent height is obtained by multiplying with the common ratio, 0.75 in this case, reflecting the ball's elasticity.
• The sequence for the ball's bounces can be calculated as: 80, 60, 45, 33.75, and so on.
• Mathematically, the height of the ball after the nth bounce can be expressed using the formula: \[ h_n = 80 \times \left( \frac{3}{4} \right)^n \] where \( h_n \) is the height of the ball on the nth bounce.

This formula provides a powerful way to predict any future bounce without having to calculate each one individually. Understanding how geometric sequences work allows us to apply them to real-world problems, like predicting rebound heights.

Elasticity

Elasticity in physics refers to the ability of an object to return to its original shape and size after being deformed, like being hit or dropped. For our bouncing ball, its elasticity directly affects how it rebounds from the ground.

• In the given problem, the elasticity is fixed such that the ball rebounds to three-fourths of the height from which it was dropped.
• Elasticity determines the common ratio in the geometric sequence for rebound heights, thus defining how much energy is conserved after each impact.
• A perfectly elastic ball would rebound to its original height, while a ball with no elasticity would not bounce at all.

The elasticity of a ball is an important factor in sports, engineering, and various practical applications where energy conservation and material performance are crucial. By understanding the elasticity of the ball, we can accurately calculate and predict the behavior of the ball over successive bounces.