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 activity features the following Vernier sensors and equipment.

### Option 2

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

## Real-World Math with Vernier

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 Mix It Up - Mixing Liquids of Different Temperatures 7 Spring Thing - Newton's Second Law 8 Stretch It to the Limit - The Linear Force Relation for a Rubber Band 9 What Goes Up - Position and Time for a Cart on a Ramp 10 That's the Way the Ball Bounces - Height and Time for a Bouncing Ball 11 Walk This Way - Definition of Rate 12 Velocity Test - Interpreting Graphs 13 From Here to There - Applications of the Distance Formula 14 Under Pressure - The Inverse Relationship between Pressure and Volume 15 Light at A Distance - Distance and Intensity 16 Chill Out: How Hot Objects Cool 17 Charging Up, Charging Down - Charging a Capacitor 18 Bounce Back - The Pattern of Rebound Heights 19 Sour Chemistry - The Exponential pH Change 20 Swinging Ellipses - Plotting an Ellipse 21 Lights Out! - Periodic Phenomena 22 Tic, Toc: Pendulum Motion 23 Stay Tuned: Sound Waveform Models 24 Up And Down: Damped Harmonic Motion 25 How Tall? Describing Data with Statistical Plots 26 And Now, the Weather - Describing Data with Statistics 27 Meet You at the Intersection: Solving a System of Linear Equations 28 Titration Curves: An Application of the Logistic Function 29 Clock Design: Period and Length of a Simple Pendulum 30 Graph It in Pieces: Piecewise Defined Functions 31 Stepping to the Greatest Integer: The Greatest Integer Function 32 Crawling Around: Parametric Plots

### Activity 18 from Real-World Math with Vernier 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.