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.

Center of mass, horizontal $0\,\mathrm{m}$

Center of mass, transverse $0\,\mathrm{m}$

Center-of-mass horizontal velocity $0\,\mathrm{m\,s^{-1}}$

Relative speed on calibrated normal $\text{No event}$

Relative speed on calibrated tangent $\text{No event}$

Camera frames through the cursor $1$

Elapsed time $0\,\mathrm{s}$

Horizontal momentum in camera frame $0\,\mathrm{kg\,m\,s^{-1}}$

Kinetic energy in camera frame $0\,\mathrm{J}$

Dissipated impact energy $0\,\mathrm{J}$

Delivered normal impulse $0\,\mathrm{N\,s}$

Predicted contact time $\text{No event}$

Restitution from camera velocities $\text{Collecting}$

Camera position residual RMS $\text{Collecting}$

Momentum conservation defect $0\,\mathrm{kg\,m\,s^{-1}}$

Kinetic plus heat balance defect $0\,\mathrm{J}$

First velocity in camera frame $0\,\mathrm{m\,s^{-1}}$

Second velocity in camera frame $0\,\mathrm{m\,s^{-1}}$

Watch a collision, then infer what happened

What survives an inelastic collision?

Past trajectoriesDraws the analytic reference path up to the cursor time.Center-of-mass markerMarks the mass-weighted average position in the chosen observer's frame.Delivered contact eventOutlines from the moment of contact and the impulse arrows appear only after the contact has happened.Velocity and contact vectorsArrows are schematic; readouts retain physical units.First mass

m_1

1.00\,\mathrm{kg}

Second mass

m_2

1.50\,\mathrm{kg}

First release velocity

u_1

1.80\,\mathrm{m\,s^{-1}}

Second release velocity

u_2

-0.40\,\mathrm{m\,s^{-1}}

Normal restitution

e

0.80\,

Initial center separation

d_0

2.40\,\mathrm{m}

Observer velocity

U

0.00\,\mathrm{m\,s^{-1}}

Camera frame rate

f_{\mathrm{cam}}

60.00\,\mathrm{Hz}

Position noise standard deviation

\sigma_{\mathrm{pos}}

0.00\,\mathrm{m}

Acquired trajectory

\tau

0.00\,\%

Drag the hollow blue release arrow, or use the sliders from the keyboard. Record collision captures the whole run with new noise; replaying or dragging through the record shows only frames already captured. Restitution estimates that fall outside the physical range are still shown, because they reveal the effect of noise.

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

BackgroundWhen 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 mattersOpenStax 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 misconceptionConserved momentum means conserved kinetic energy.

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

Sticking together needs zero restitution

Typical misconceptionEvery dissipative collision makes the carts stick.

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

Too few frames limit the measurement

Typical misconceptionIf the motion is known, the camera can always estimate it.

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

Run the experiment

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.
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.
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.
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.
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.
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.

Core mathematical model

Momentum and impulse

Restitution from two velocity fits

Energy lost in the impact

Changing the observer

Common difficulties

Momentum is not energy

Sticking together needs zero restitution

Too few frames limit the measurement

Run the experiment

Find the instant of impact

Follow the center of mass

Measure before comparing

Exchange or share velocity

Move the observer

Push the measurement until it fails