M006 · Observers & invariants

Galilean Frame Switcher

Which quantities of motion change when the observer changes? Coordinates, momentum and energy depend on the observer; relative motion stays the same. Kinetic energy is least in the mass-centre frame.

Switch between platform, train and mass-centre observers. Drag the gliders, compare relative motion and find the observer with the least kinetic energy.

Nearest valid window time $0$

Estimated horizontal momentum $0$

Estimated kinetic energy $0$

Motion state $0$

Selected time $0$

Observer horizontal speed $0$

Horizontal velocity of A $1\,\mathrm{m\,s^{-1}}$

A transverse velocity $0$

Horizontal velocity of B $-1\,\mathrm{m\,s^{-1}}$

B transverse velocity $0$

Total horizontal momentum $-1\,\mathrm{kg\,m\,s^{-1}}$

Total transverse momentum $0$

Frame kinetic energy $1.5\,\mathrm{J}$

Relative-motion energy $1.33\,\mathrm{J}$

Mass-centre kinetic energy in frame $0$

Energy lost in impact $0\,\mathrm{J}$

Energy decomposition residual $0$

Frame transformation residual $0$

Raw position frames $0$

Available slope estimates $0$

One motion, different observers

Can a moving body stand still?

Mass of A

m_A

1.00\,kg

Mass of B

m_B

2.00\,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

4.00\,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}

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

0.35\,

Two ideal gliders ride on separate parallel tracks and do not interact. Each keeps its laboratory velocity, and the tracks never pass momentum from one glider to the other. The experiment shows the zero-acceleration case of Galilean invariance. Gravity and the support force cancel vertically; air resistance, propulsion and the ends of the tracks are left out.

Pick an observer with the buttons above the apparatus or with the observer slider. All coordinate origins coincide at the start, although the mass centre need not start at the origin. Every observer moves at a constant horizontal velocity. “Train” and “conveyor” are only names for observers, not moving surfaces that exert forces. Changing observer changes the coordinates and the energies each observer computes, never the physical experiment.

The two horizontal coordinates lie in the table plane, and the vertical axis points up. Glider A is gold and glider B is blue. One metre is drawn the same length in every direction. The green cross marks the geometric mass centre; it is not a third body. Velocity arrows use a fixed scale. The motion replays at half speed, or drag the time slider. The default camera keeps both bodies in view; playback pauses when the apparatus is off screen or the tab is hidden.

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

Velocities come from centred least-squares slopes. Each slope needs a complete window of evenly timed samples, so there are no estimates at the ends of the record or across gaps. A window that spans an impact gives an average across the change in velocity, not the velocity just before or just after the impact. The estimate readouts use the valid window nearest the clock and show its centre time. The analysis reads only the recorded positions, the given masses and the observer velocity; it does not read the ideal collision time. A wider window reduces noise but also blurs sudden changes.

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

Model assumptions

Two noninteracting gliders follow parallel ideal guides; observers move at constant horizontal velocities.

Start with the essentials

Focus question: Which observer sees A at rest, and which sees zero total momentum?
One-sentence intuition: Riding alongside one glider removes its velocity from your view. Riding with the mass centre removes the momentum of the whole system; these are generally different choices.

Core mathematical model

Translate the coordinates

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

The coordinate origins coincide at the initial time. The observer moves horizontally at a constant velocity; time and the transverse coordinate stay the same.

Add both momenta

\begin{aligned}P_x^\prime&=P_x-MU,\\ P_y^\prime&=P_y.\end{aligned}

Total mass is the sum of the two glider masses. Subtracting the observer velocity from each body changes the total momentum by total mass times that velocity.

Find the energy minimum

\begin{aligned}K^\prime&=K-UP_x+\tfrac12MU^2,\\ K^\prime&=K_{\mathrm{CM}}^\prime+K_{\mathrm{rel}},\\ K_{\mathrm{CM}}^\prime&=\frac{\lVert\mathbf P^\prime\rVert^2}{2M}.\end{aligned}

The relative-motion energy uses velocities measured from the mass centre. It stays when the momentum of the whole system is zero. This minimum does not select an absolute rest frame.

Compare one glider with the other

\begin{aligned}\mathbf r_{BA}^\prime&=\mathbf r_{BA},\\ \mathbf v_{BA}^\prime&=\mathbf v_{BA}.\end{aligned}

The subscript means the position or velocity of B minus that of A. The observer correction cancels. Parallel tracks keep the gliders apart even when their horizontal coordinates coincide.

Common difficulties

Being still beside A is not the same as following the whole system

Typical misconceptionIf A stands still, the total momentum and kinetic energy must both vanish.

Better mental modelB can still move in that frame. The mass-centre observer balances both momenta. Its kinetic energy vanishes only when there is no relative motion.

Run the experiment

01
Ride alongside one glider
Choose Ride alongside A, then switch between Platform and Train.
What to observe: A’s velocity arrow disappears in the matching train frame. The laboratory ruler moves past it.
02
Find the lowest energy
Drag across the observer-energy curve, then choose Mass centre.
What to observe: The marker reaches the minimum when the total momentum vanishes. The relative-motion part of the energy bar usually remains.
03
Keep the same position samples
Open Position recording and inference, record positions and switch observers.
What to observe: The raw laboratory rows and their error are unchanged. Estimated velocities subtract the new observer velocity from slopes of those same rows.

Physical model, position data and sources

The two physical coordinates are horizontal; the vertical axis points up. Position is in metres, velocity in metres per second, mass in kilograms and energy in joules. Observer origins coincide at the initial time and move at constant horizontal velocities. Train and conveyor name observers; they exert no drive or friction on the bodies.

The gliders do not interact and follow separate parallel ideal guides. Gravity and vertical support balance; their horizontal accelerations are zero. Propulsion, air resistance and collisions with the track ends are omitted.

All apparatus lengths share one scale. The camera frames the selected run and can change its framing between observers; the numerical coordinates and ruler set the length reference. Velocity arrows start above the bodies for visibility and show one quarter-second of travel. The green cross is the mass-weighted centre, not a third particle.

On entry, position recording loads a simulated run. The CSV contains only timestamps, frame indices and laboratory positions with deterministic bounded coordinate error. Its noise is not a calibrated confidence model. Model curves and recorded dots are labelled separately. Changing the motion, masses or recording settings clears the rows; changing observer, time or slope window preserves them.

The inference uses only those rows, supplied masses, observer velocity and an odd centred least-squares window. Endpoints and gaps have no estimates. A window spanning contact averages the velocity jump and can underestimate instantaneous kinetic energy. The estimator never reads the model event time; it reports the time of the window’s centre sample.