M031 · Observers & invariants

Center-of-Momentum Frame Lab

Resolve an oblique two-body impact, switch observers and separate mass-centre motion from internal kinetic energy. Compare elastic exchange, 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 / ACQUIRE / TRANSFORM

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, equal-radius spheres move on a frictionless horizontal plane. The initial velocity is horizontal; the offset selects an oblique contact or a miss. The normal impulse follows constant restitution; tangential relative velocity is unchanged. No friction, spin, deformation, finite contact duration or external horizontal force is modeled. Gravity and support cancel vertically.

Zero restitution removes only relative normal motion. An oblique encounter can retain tangential motion and separate; it does not glue the spheres together. The head-on zero-restitution preset leaves them touching at a common velocity. The event time and total impulse are model references, not measured peaks.

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: Can total momentum vanish while both bodies keep moving?
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.

Resolve the normal impulse

\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}

Relative velocity is B minus A; the normal points from A to B. Only closing contact receives an impulse.

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
Resolve the encounter
Compare oblique, elastic, zero-restitution and miss presets.
What to observe: Impulse and dissipated energy are observer invariant.
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

Resolve the normal impulse

Common difficulties

Separate frame choice from dynamics

Run the experiment

Predict the observer

Move along the energy curve

Resolve the encounter

Acquire before switching