---
title: "One-Dimensional Collision Bench Interactive Lab | Sandbox Physics"
description: "What survives an inelastic collision? Total momentum survives; kinetic energy does not. Observers moving at different steady speeds disagree about each momentum, but they all agree on the restitution coefficient and on how much energy was lost, because both depend only on relative velocity."
canonical_url: "https://sandboxphysics.com/labs/collision-bench/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/collision-bench/"
last_modified: "2026-10-22"
---

M032 · Momentum / restitution

# One-Dimensional Collision Bench

**What survives an inelastic collision?** Total momentum survives; kinetic energy does not. Observers moving at different steady speeds disagree about each momentum, but they all agree on the restitution coefficient and on how much energy was lost, because both depend only on relative velocity.

Release two carts on a frictionless rail and turn the three-dimensional view to watch from any side. Compare an elastic exchange, a lossy impact and carts that end with a common velocity, and fit the camera track before and after the contact trigger. A moving observer measures different momentum and kinetic energy, but the same restitution and the same energy lost in the collision.

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

The carts are smooth point masses with fixed contact radii that are drawn only for illustration. There is no friction, no spin and no external horizontal impulse. The motion and the contact time are solved analytically. The camera adds independent Gaussian noise to each position, and its time and length scales are calibrated. The contact timing is calibrated separately. The camera segments before and after impact each need at least four frames.

Physics tutorial

## A collision seen by two observers

**Background** When two carts collide, each gives the other an equal and opposite impulse. Total momentum is conserved in every inertial frame, but kinetic energy is conserved only in an elastic collision.

**Why it matters** OpenStax University Physics, sections 9.4 and 9.5, covers momentum and collisions. This Lab adds a calibrated contact trigger and a noisy camera, so you have to measure restitution from two fitted velocity segments, one before impact and one after.

Start with the essentials

- **Focus question**: Can a moving camera change the energy loss or the restitution of the same impact?
- **One-sentence intuition**: Fit the velocities first, then compare the relative velocities along the normal before and after impact. The observer's own velocity cancels when you subtract one cart's velocity from the other's.

### Core mathematical model

#### Momentum and impulse

$$
m_1u_1+m_2u_2=m_1v_1+m_2v_2,\qquad m_1(v_1-u_1)=-m_2(v_2-u_2)
$$

There is no external horizontal impulse during contact.

#### Restitution from two velocity fits

$$
e=\frac{v_2-v_1}{u_1-u_2},\qquad u_1>u_2
$$

Each side needs at least four frames. If the approach speed is poorly resolved, the ratio is unreliable.

#### Energy lost in the impact

$$
K_{\mathrm{before}}-K_{\mathrm{after}}=\frac12\frac{m_1m_2}{m_1+m_2}(1-e^2)(u_1-u_2)^2
$$

The Lab recomputes the energy from both final velocities and compares the loss with this reduced-mass formula.

#### Changing the observer

$$
v_i'=v_i-U,\quad P'=P-(m_1+m_2)U,\quad K'=K-UP+\frac12(m_1+m_2)U^2
$$

Because momentum is conserved, the terms that depend on the observer's velocity cancel when you subtract the kinetic energy after impact from the energy before.

### Common difficulties

#### Momentum is not energy

**Typical misconception** Conserved momentum means conserved kinetic energy.

**Better mental model** In a lossy impact, momentum is still conserved, but part of the kinetic energy is lost in the collision.

#### Sticking together needs zero restitution

**Typical misconception** Every dissipative collision makes the carts stick.

**Better mental model** On this one-dimensional rail, the carts move off together only when the restitution is zero.

#### Too few frames limit the measurement

**Typical misconception** If the motion is known, the camera can always estimate it.

**Better mental model** With too few frames before impact, the camera gives no fit, even though the model still predicts the impact.

### Run the experiment

1. 01

   #### Find the instant of impact

   Step through Just before impact, At impact and Just after impact, rotate the view, then switch to Conservation ledger.

   **What to observe:** Positions change smoothly, but the velocities jump. The violet arrows show impulses already delivered at impact, not forces acting throughout the playback.

2. 02

   #### Follow the center of mass

   Set unequal masses and watch the teal center-of-mass marker as you drag through the record.

   **What to observe:** Its velocity stays constant even when the carts' velocities change. The momentum and energy totals cover only the time up to the cursor.

3. 03

   #### Measure before comparing

   Record the default impact and compare the measured restitution with its setting.

   **What to observe:** The camera dots and fitted velocities are noisy; the analytic trajectory serves as the reference.

4. 04

   #### Exchange or share velocity

   Compare the Equal-mass exchange and Common final velocity presets.

   **What to observe:** Only the common-velocity case removes all rail-relative kinetic energy.

5. 05

   #### Move the observer

   After recording, change Observer velocity while keeping the same cursor and camera samples.

   **What to observe:** Momentum and kinetic energy change, but the energy lost in the impact and the restitution stay the same.

6. 06

   #### Push the measurement until it fails

   Try the Moving apart and Too few pre-impact frames presets.

   **What to observe:** Carts that never meet receive no impulse, and too few frames give no fit; the Lab reports these as different failures.

Physical conclusion

Momentum depends on the observer. The kinetic energy lost in an isolated impact does not, and neither does the restitution, which uses only relative velocities.

Starter path · Step 4 of 6

## From a falling ball to an orbit

[Next step → **Mass–Spring Oscillator** Change the mass, stiffness and release distance, then measure eight periods to find which one actually sets the period. Model · 18 min](https://sandboxphysics.com/labs/mass-spring/)

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)