M006 · Observers & invariants

Galilean Frame Switcher

Move between platform, train, conveyor and mass-centre observers. Keep the same two gliders and recorded positions while comparing momentum and the kinetic-energy minimum.

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 $\text{Ready}$

OBSERVE / ACQUIRE / TRANSFORM

Keep the experiment, change the observer

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}

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

0.35\,

Two noninteracting ideal gliders occupy separate parallel tracks. Each keeps its laboratory velocity; the tracks never exchange momentum between gliders. This experiment demonstrates the zero-acceleration limit of Galilean invariance. Gravity and support cancel vertically; air resistance, propulsion and track ends are omitted.

Choose an observer above the apparatus or use the observer slider. All coordinate origins coincide initially; the mass centre need not start at the origin; every observer has constant horizontal velocity. Train and conveyor are observer names, not moving surfaces that exert forces. Changing observers changes coordinates and energy bookkeeping, never the physical experiment.

Physical horizontal coordinates lie in the table plane; the vertical axis points upward. Gold is A, blue is B. One metre has one scale in every direction. The green cross marks the geometric mass centre; it is not another body. Arrows show velocity with a fixed gain. Replay is manual, with a draggable gold clock and keyboard sliders.

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

Centred least-squares slopes use complete uniformly timed windows. They omit endpoints and gaps. A window spanning an impact reports an averaged velocity transition, not instantaneous pre-impact or post-impact velocity. Estimate readouts show the valid window nearest the clock and its centre time. The analysis reads recorded positions, supplied masses and observer velocity; it does not read the ideal collision time. Wider windows reduce noise and blur transitions.

Raw CSV contains every position row in laboratory coordinates. The table previews all rows up to 160, otherwise the first and last 80. Reconstructed momentum and kinetic energy depend on the chosen masses and noisy estimated velocities. They are estimates, distinct from model reference curves. Controls, plots and export remain available without WebGL.

Ideal Newtonian mechanics with explicit acquisition boundaries.

Start with the essentials

Focus question: Whose motion changes when the observer changes?
One-sentence intuition: Zero total momentum minimizes kinetic energy without eliminating internal motion.

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}

Only constant horizontal boosts are used. Time and transverse coordinates are unchanged.

Momentum changes with the observer

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

Total momentum vanishes in the centre frame for this initial condition.

Energy has a minimum

\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 residual is checked against the direct sum of the two body energies.

What survives a boost?

\begin{aligned}\mathbf r_B^\prime-\mathbf r_A^\prime&=\mathbf r_B-\mathbf r_A\\\mathbf v_B^\prime-\mathbf v_A^\prime&=\mathbf v_B-\mathbf v_A\end{aligned}

Separations and relative velocities remain the same; both accelerations are zero here.

Common difficulties

Separate frame choice from dynamics

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

Better mental modelThe same physical positions and impulses receive different coordinates. Internal kinetic energy persists unless the 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: One stationary body does not generally make total momentum vanish.
02
Move along the energy curve
Drag the copper handle across the observer-energy plot.
What to observe: The minimum follows the total mass and momentum, not a preferred absolute rest frame.
03
Keep both tracks
Compare same-velocity and opposite-velocity gliders.
What to observe: Passing positions on separate tracks do not cause a collision.
04
Acquire before switching
Acquire positions, change observers and compare finite-window estimates with the reference.
What to observe: The raw laboratory record is identical after a frame switch.

Core mathematical model

The same event in another frame

Momentum changes with the observer

Energy has a minimum

What survives a boost?

Common difficulties

Separate frame choice from dynamics

Run the experiment

Predict the observer

Move along the energy curve

Keep both tracks

Acquire before switching