---
title: "Center-of-Momentum Frame Lab Interactive Lab | Sandbox Physics"
description: "Can total momentum vanish while both bodies keep moving? Yes."
canonical_url: "https://sandboxphysics.com/labs/centre-momentum/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/centre-momentum/"
last_modified: "2026-10-27"
---

M031 · Observers & invariants

# Center-of-Momentum Frame Lab

**Can total momentum vanish while both bodies keep moving?** Yes. In the center-of-momentum frame the two bodies have equal and opposite momenta, so the total is zero while both still move. This frame also has the least kinetic energy, and that leftover energy is the most a collision can lose.

Watch two spheres collide at an angle, then view the collision from different moving observers. Split the kinetic energy into mass-centre motion and internal motion, and compare an elastic collision, a collision with normal restitution and a clean miss.

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

The Lab uses ideal Newtonian mechanics and states clearly what the recorded data can and cannot tell you.

Physics tutorial

## Center-of-Momentum Frame Lab

**Background** In Newtonian mechanics, observers moving at different constant velocities are all equally valid.

**Why it matters** Different observers see different momenta and kinetic energies in the same collision. Sorting out which quantities change and which stay the same shows what the collision itself does.

Start with the essentials

- **Focus question**: Can total momentum vanish while both bodies keep moving?
- **One-sentence intuition**: The observer who sees zero total momentum measures the least kinetic energy, yet the bodies still move relative to each other.

### Core mathematical model

#### The same event in another frame

$$
\begin{aligned}x_i^\prime&=x_i-Ut\\v_{x,i}^\prime&=v_{x,i}-U\end{aligned}
$$

Every observer moves at a constant horizontal velocity. Time and the transverse coordinate stay the same.

#### Momentum changes with the observer

$$
P_x^\prime=P_x-MU,\qquad P_y^\prime=P_y
$$

For these initial conditions the total momentum is zero in the mass-centre frame.

#### Kinetic energy is lowest in the mass-centre frame

$$
\begin{aligned}K^\prime&=K-UP_x+\tfrac12MU^2\\&=\tfrac12M|\mathbf V_{\rm cm}-U\hat{\mathbf x}|^2+K_{\rm int}\end{aligned}
$$

The Lab compares this formula with the kinetic energies of the two bodies added directly; the residual is the difference.

#### The normal impulse and the energy lost

$$
\begin{aligned}J&=-(1+e)\mu\,\mathbf w^-\!\cdot\mathbf n\\Q&=\tfrac12\mu(1-e^2)(\mathbf w^-\!\cdot\mathbf n)^2\end{aligned}
$$

The relative velocity is B's velocity minus A's, and the normal points from A to B. An impulse acts only when the spheres are moving toward each other at contact.

### Common difficulties

#### Changing observer does not change the motion

**Typical misconception** Changing the observer changes the collision or removes all kinetic energy.

**Better mental model** The same positions and impulses are simply described with different coordinates. Internal kinetic energy remains unless an interaction dissipates it.

### Run the experiment

1. 01

   #### Predict the observer

   Find an observer that makes A stationary, then compare the mass-centre observer.

   **What to observe:** Making one body stationary does not, in general, make the total momentum zero.

2. 02

   #### Move along the energy curve

   Drag the copper handle across the observer-energy plot.

   **What to observe:** The minimum is set by the total mass and momentum; it does not mark an absolute rest frame.

3. 03

   #### Compare the encounters

   Compare oblique, elastic, zero-restitution and miss presets.

   **What to observe:** Every observer finds the same impulse and the same dissipated energy.

4. 04

   #### Acquire before switching

   Acquire positions, change observers and compare finite-window estimates with the reference.

   **What to observe:** After you switch observers, the raw laboratory record is exactly the same.

Physical conclusion

Momentum and kinetic energy change from one observer to another; relative motion and the energy lost in a collision do not.

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)