Vernier Software & Technology

# That's the Way the Ball Bounces - Height and Time for a Bouncing Ball

## Introduction

Picture a bouncing ball. Between impacts with the floor, the ball rises and slows, then descends and speeds up. For any particular bounce, if the ball’s height is plotted as a function of time, the resulting graph has a parabolic shape. In other words, the relationship between height and time for a single bounce of a ball is quadratic. This relationship is expressed mathematically as

$y = a{x^2} + bx + c$

where y represents the ball’s height at any given time x. Another form of a quadratic equation is

$y = {a{(x - h)^2} + k}$

where h is the x-coordinate of the vertex, k is the y-coordinate of the vertex, and a is a parameter. This way of writing a quadratic is called the vertex form.

In this activity, you will record the motion of a bouncing ball using a Motion Detector. You will then analyze the collected data and model the variations in the ball’s height as a function of time during one bounce using both the general and vertex forms of the quadratic equation.

## Objectives

• Record height versus time data for a bouncing ball.
• Model a single bounce using both the general and vertex forms of the parabola.

## 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 11 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.