---
title: "Bouncing Ball & Restitution Interactive Lab | Sandbox Physics"
description: "Where does the missing height go? It is lost in each impact."
canonical_url: "https://sandboxphysics.com/labs/bouncing-ball/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/bouncing-ball/"
last_modified: "2026-10-27"
---

M050 · Every bounce tells a story

# Bouncing Ball & Restitution

**Where does the missing height go?** It is lost in each impact. The coefficient of restitution is the rebound speed divided by the impact speed, and the height a ball reaches grows with its speed squared. Each new peak is the previous one times the coefficient squared, so a ball with a coefficient of 0.9 keeps 81 percent of its height on every bounce.

Drop a ball, watch the impact in slow motion and compare elastic, lossy and sticking collisions. Record apex heights to estimate the coefficient of restitution, and see how the ideal model fits infinitely many bounces into a finite time.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Impacts are instantaneous and the floor does not move. The ideal model's infinite run of bounces in a finite time marks the edge of the model.

Physics tutorial

## Bouncing Ball & Restitution

**Background** A fixed floor exchanges momentum with a ball at each impact.

**Why it matters** Each bounce is lower than the last, and the lost height tells you how much mechanical energy left the ball.

Start with the essentials

- **Focus question**: Where does the missing height go?
- **One-sentence intuition**: The ratio of successive bounce heights is the square of the speed ratio.

### Core mathematical model

#### Instantaneous impact

$$
v^+=-e v^-,\qquad J=m(v^+-v^-)
$$

The floor takes the opposite impulse; the ball alone does not conserve momentum.

#### Apex height and energy after each bounce

$$
H_n=e^{2n}H_0,\qquad E_n=mgH_n
$$

Uniform gravity and no drag are assumed between instantaneous impacts.

#### Infinitely many bounces in a finite time

$$
t_*=\sqrt{\frac{2H_0}{g}}+\frac{2e\sqrt{2gH_0}}{g(1-e)}
$$

With zero restitution the first impact stops the ball. With a restitution of one, the bounces never die out.

#### Restitution from recorded heights

$$
\widehat e=\sqrt{\frac{\sum_n H_nH_{n+1}}{\sum_n H_n^2}}
$$

The fit is a line through the origin, using resolved pairs of successive recorded heights. Noise in both heights of a pair can bias the estimate.

### Common difficulties

#### The floor pushes from outside

**Typical misconception** The ball conserves momentum when it hits a fixed floor.

**Better mental model** The floor gives the ball an external impulse, so the ball's momentum changes. Count the energy transferred at impact as a separate term.

### Run the experiment

1. 01

   #### Set the height

   Drag the gold handle or use its keyboard slider.

   **What to observe:** The scene and predicted motion update together.

2. 02

   #### Follow the event

   Play, pause and jump to the next event.

   **What to observe:** Height changes smoothly, but an impact can change the velocity instantly.

3. 03

   #### Track the energy

   Drag the time between turning points.

   **What to observe:** Stored, kinetic, gravitational and transferred energy always add up to the same total.

4. 04

   #### Measure a run

   Record apex heights, export the CSV and compare the inferred parameter.

   **What to observe:** Replaying preserves the same measurements; changing physical parameters clears them.

Physical conclusion

The ball loses energy, yet the total still balances once you count the energy transferred at impacts.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)