 Vernier Software & Technology

# Bounce Back - The Pattern of Rebound Heights

## Introduction

When a ball bounces up and down on a flat surface, the maximum height it reaches decreases from bounce to bounce. In fact, the maximum height decreases in a very predictable way for most types of balls. The relationship between the maximum height attained by the ball on a given bounce (which we will call the rebound height) and number of bounces that have occurred since the ball was released is an exponential $y = h{p^x}$

where y represents the rebound height, x represents the bounce number, h is the release height, and p is a constant that depends on the physical characteristics of the ball used. It’s easy to see where this model comes from: Suppose that the ball is released from height h. Then on each bounce it rebounds to a fraction p of the previous maximum height. After zero, one and two bounces, the ball will attain a maximum height of h, hp, (hp)p = hp2, and so forth. The relation above is generalized for any x number of bounces.

In this exercise, you will collect motion data for a bouncing ball using a Motion Detector. You will then analyze this data to test the model y = hpx.

## Objectives

• Record the successive maximum heights for a bouncing ball.
• Model the bounce height data with an exponential function.

## Sensors and Equipment

This experiment features the following Vernier sensors and equipment.

### Additional Requirements

You may also need an interface and software for data collection. What do I need for data collection?

## Real-World Math Made Easy

See other experiments from the lab book.

 1 Walk the Line - Straight Line Distance Graphs 2 Making Cents of Math: Linear Relationship between Weight and Quantity 3 Pool Plunge - Linear Relationship between Water Depth and Pressure 4 Funnel Volumes - Volume and Weight 5 Keep It Bottled Up - Rates of Pressure Increase 6 Graph It in Pieces: Piecewise Defined Functions 7 Mix It Up - Mixing Liquids of Different Temperatures 8 Spring Thing - Newton's Second Law 9 Stretch It to the Limit - The Linear Force Relation for a Rubber Band 10 What Goes Up - Position and Time for a Cart on a Ramp 11 That's the Way the Ball Bounces - Height and Time for a Bouncing Ball 12 Walk This Way - Definition of Rate 13 Velocity Test - Interpreting Graphs 14 From Here to There - Applications of the Distance Formula 15 Under Pressure - The Inverse Relationship between Pressure and Volume 16 Light at A Distance - Distance and Intensity 17 Chill Out: How Hot Objects Cool 18 Charging Up, Charging Down - Charging a Capacitor 19 Bounce Back - The Pattern of Rebound Heights 20 Sour Chemistry - The Exponential pH Change 21 Stepping to the Greatest Integer: The Greatest Integer Function 22 Swinging Ellipses - Plotting an Ellipse 23 Crawling Around: Parametric Plots 24 Lights Out! - Periodic Phenomena 25 Tic, Toc: Pendulum Motion 26 Stay Tuned: Sound Waveform Models 27 Up And Down: Damped Harmonic Motion 28 How Tall? Describing Data with Statistical Plots 29 And Now, the Weather - Describing Data with Statistics 30 Meet You at the Intersection: Solving a System of Linear Equations 31 Titration Curves: An Application of the Logistic Function

### Experiment 19 from Real-World Math Made Easy Lab Book #### Included in the Lab Book

Vernier lab books include word-processing files of the student instructions, essential teacher information, suggested answers, sample data and graphs, and more.