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.

Nearest valid window time $\text{Ready}$

Estimated horizontal momentum $\text{Ready}$

Estimated kinetic energy $\text{Ready}$

Motion state $\text{Ready}$

Selected time $\text{Ready}$

Observer horizontal speed $\text{Ready}$

A horizontal velocity in frame $\text{Ready}$

A transverse velocity $\text{Ready}$

B horizontal velocity in frame $\text{Ready}$

B transverse velocity $\text{Ready}$

Total horizontal momentum in frame $\text{Ready}$

Total transverse momentum $\text{Ready}$

Total kinetic energy in frame $\text{Ready}$

Internal kinetic energy $\text{Ready}$

Mass-centre kinetic energy in frame $\text{Ready}$

Dissipated energy up to clock $\text{Ready}$

Energy decomposition residual $\text{Ready}$

Frame transformation residual $\text{Ready}$

Raw position frames $\text{Ready}$

Available slope estimates $\text{Ready}$

Model contact time $\text{Ready}$

Impulse up to clock $\text{Ready}$

Observe, record, then change the frame

Keep the experiment, change the observer

Mass of A

m_A

1.00\,kg

Mass of B

m_B

1.00\,kg

Initial horizontal velocity of A

v_{A,0}

3.00\,\frac{\mathrm m}{\mathrm s}

Initial horizontal velocity of B

v_{B,0}

-1.00\,\frac{\mathrm m}{\mathrm s}

Run duration

T

3.00\,s

Observer selection

S

0.00\,

Train observer velocity

U_{\mathrm{train}}

2.00\,\frac{\mathrm m}{\mathrm s}

Conveyor observer velocity

U_{\mathrm{belt}}

-1.00\,\frac{\mathrm m}{\mathrm s}

Custom observer velocity

U

0.00\,\frac{\mathrm m}{\mathrm s}

Position sampling rate

f_s

10.00\,Hz

Bounded position error

\epsilon

0.01\,m

Deterministic error seed

s

23.00\,

Centred slope window

w

5.00\,

Replay fraction

\tau

0.60\,

Transverse centre offset

b

0.35\,m

Normal restitution

e

0.80\,

Two smooth spheres of equal radius move on a frictionless horizontal plane. Both start moving along the horizontal axis, and the offset decides between an oblique contact and a miss. The normal impulse follows a constant coefficient of restitution, and the tangential relative velocity does not change. Friction, spin, deformation, a finite contact time and external horizontal forces are not modeled. Gravity and the support force cancel vertically.

Zero restitution removes only the relative motion along the normal. In an oblique encounter the spheres can keep their tangential motion and separate; they do not stick together. The head-on zero-restitution preset leaves them touching and moving at a common velocity. The event time and total impulse are model references, not measured peaks.

Pick an observer with the buttons above the apparatus or with the observer slider. All coordinate origins start at the same point, the mass centre need not start at the origin, and every observer moves at a constant horizontal velocity. Train and conveyor are only names for observers, not moving surfaces that exert forces. Changing the observer changes the coordinates and how the energy is divided up, never the physical experiment.

Both physical coordinate axes lie in the table plane, and the vertical axis points up. Gold is A and blue is B. One metre has the same scale in every direction. The green cross marks the geometric mass centre; it is not another body. Arrows show velocity at a fixed gain. Click Play for a half-speed replay, or use the time slider. The default camera follows both bodies, and playback pauses when the view is offscreen or the tab is hidden.

Acquire stores only sampled laboratory positions and their timestamps, with a bounded, deterministic added error. The noise is not a calibrated confidence model. Switching observers, replaying or changing the slope window keeps the same raw record; changing the masses, motion or sensor settings clears it.

Velocities come from centred least-squares slopes over complete, evenly timed windows, so endpoints and gaps are left out. A window that spans the impact gives an averaged change in velocity, not the instantaneous velocity just before or after impact. The estimate readouts show the valid window nearest the clock and its centre time. The analysis uses only the recorded positions, the given masses and the observer velocity; it never reads the ideal collision time. Wider windows reduce noise but also blur sudden changes.

The raw CSV contains every position row in laboratory coordinates. The table shows every row when there are at most 160, and otherwise the first and last 80. Reconstructed momentum and kinetic energy depend on the chosen masses and the noisy velocity estimates. They are estimates, separate from the model reference curves. Controls, plots and export still work without WebGL.

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

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 misconceptionChanging the observer changes the collision or removes all kinetic energy.

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

Run the experiment

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.
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.
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.
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.

Core mathematical model

The same event in another frame

Momentum changes with the observer

Kinetic energy is lowest in the mass-centre frame

The normal impulse and the energy lost

Common difficulties

Changing observer does not change the motion

Run the experiment

Predict the observer

Move along the energy curve

Compare the encounters

Acquire before switching