Calculating the Total Distance Traveled by a Falling Rubber Ball

Imagine a rubber ball that is dropped from a height of 16 feet. At each rebound, the ball rises to a height that is 3/4 of the height of the previous fall. How can we calculate the total distance the ball will have traveled before it comes to rest?

Understanding the Motion and Applying Geometric Series

When a ball is dropped, it travels a certain distance downward. Upon rebound, it travels back upward to a height that is 3/4 of the previous height. This rebound process continues, each time reducing the height of the fall and the rise by a factor of 3/4. This scenario uses the concept of a geometric series, where each term is obtained by multiplying the previous term by a constant factor (in this case, 3/4).

Total Distance Calculation

First Drop

The ball initially falls from a height of 16 feet, traveling 16 feet downward. Let's consider the subsequent falls and rises:

For the first fall, ( S_1 16 ) feet.

Subsequent Falls and Rises

At the first rebound, the ball rises to a height of ( frac{3}{4} times 16 12 ) feet and then falls back down 12 feet, totaling ( 2 times 12 24 ) feet.

At the second rebound, the ball rises to ( frac{3}{4} times 12 9 ) feet and falls back 9 feet, totaling ( 2 times 9 18 ) feet. This process repeats, each time the height and the distance traveled by the ball being multiplied by ( frac{3}{4} ).

Formulating the Total Distance

Let’s calculate the total distance:

Total distance for the first fall: ( 16 ) feet Total distance for the first rise and fall: ( 2 times 12 24 ) feet Total distance for the second rise and fall: ( 2 times 9 18 ) feet Total distance for the third rise and fall: ( 2 times 6.75 13.5 ) feet, and so on.

The total distance traveled can be calculated using the formula for the sum of an infinite geometric series:

Total distance ( S ) Initial distance down Sum of distances of rises and falls

The sum of the series for the rise and fall distances is given by:

[ S 16 2 times left(12 9 6.75 ldots right) ]

The series inside the parentheses is a geometric series with the first term ( a 12 ) and common ratio ( r frac{3}{4} ).

The sum of the infinite geometric series is:

[ text{Sum} frac{a}{1 - r} frac{12}{1 - frac{3}{4}} frac{12}{frac{1}{4}} 48 text{ feet} ]

Therefore, the total distance traveled is:

[ S 16 2 times 48 16 96 112 text{ feet} ]

Examples and Calculations

Let's look at some specific calculations to confirm the total distance.

First Example:

Given the initial height of 16 feet and the factor of ( frac{3}{4} ), we can calculate the total distance using the formula:

[ text{Total distance} 16 2 times left(12 9 6.75 ldots right) 112 text{ feet} ]

Second Example:

Considering a different initial height of 12 feet and the same factor of ( frac{3}{4} ), we can calculate the total distance as:

[ text{Total distance} 12 2 times left(9 6.75 5.06 ldots right) 84 text{ feet} ]

Third Example:

For a series with an initial height of 12 feet and a different factor of ( frac{2}{3} ), we can calculate the total distance as:

[ text{Total distance} 12 2 times left(8 5.33 3.55 ldots right) 60 text{ feet} ]

Conclusion

The total distance traveled by the falling rubber ball before it finally comes to rest can be calculated using the concept of geometric series. In the scenarios described, the total distances are:

112 feet for a 16-foot drop with a rise of ( frac{3}{4} ) of the previous height. 84 feet for a 12-foot drop with a rise of ( frac{3}{4} ) of the previous height. 60 feet for a 12-foot drop with a rise of ( frac{2}{3} ) of the previous height.

This article demonstrates the application of geometric series in solving real-world problems, particularly in physics and mechanics.