### 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.