M032 · Momentum / restitution

One-Dimensional Collision Bench

Release two carts on a frictionless rail. Compare elastic exchange, dissipation and a common final velocity; fit camera tracks before and after the contact trigger. A moving observer changes momentum and kinetic energy while leaving restitution and collision loss unchanged.

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 $0\,\mathrm{m\,s^{-1}}$

RELEASE / OBSERVE / INFER

What survives an inelastic collision?

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 keyboard sliders. Record collision collects the entire run with fresh noise; replay and scrub reveal only acquired frames. Estimates outside the physical restitution interval remain visible as noise diagnostics.

Smooth point-mass carts with fixed schematic contact radii, no friction, spin or external horizontal impulse. Motion and contact time are solved analytically. Camera noise is independent Gaussian position noise with calibrated time and length. Contact timing is independently calibrated. Both camera segments require at least four frames.

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. Compare the two normal relative velocities; an observer boost cancels from their difference.

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

Four frames on each side are the minimum. A poorly resolved closing velocity makes the ratio unreliable.

An independent loss identity

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

The energy ledger is recomputed from both final velocities, then compared with this reduced-mass reference.

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

The conserved momentum makes the boost terms cancel from the before–after kinetic energy difference.

Common difficulties

Momentum is not energy

Typical misconceptionConserved momentum means conserved kinetic energy.

Better mental modelUse the dissipative preset: momentum closes while kinetic energy becomes impact loss.

A common velocity is special

Typical misconceptionEvery dissipative collision makes the carts stick.

Better mental modelOnly zero restitution removes the relative normal velocity on this one-dimensional rail.

Sampling is an instrument limit

Typical misconceptionA known trajectory guarantees a camera estimate.

Better mental modelChoose too few pre-impact frames; the instrument refuses a fit even though the model predicts the impact.

Run the experiment

01
Measure before comparing
Record the default impact and compare the measured restitution with its setting.
What to observe: Dots and fitted velocities are noisy; the analytic trajectory is the reference.
02
Exchange or share velocity
Compare equal-mass elastic and common-velocity presets.
What to observe: Only the common-velocity case removes all rail-relative kinetic energy.
03
Move the observer
Change observer velocity and record again.
What to observe: Momentum and kinetic energy change; impact loss and restitution agree.
04
Break the instrument
Try separating carts and too few pre-impact frames.
What to observe: No impulse and insufficient sampling produce different failure states.