M041 · Work and energy

Work–Energy Graph Studio

Draw the force along a rail, follow exact piecewise motion through turns, then integrate acquired force samples against the independent energy ledger.

Trajectory window $\text{Ready}$

Selected time $\text{Ready}$

Selected position $\text{Ready}$

Signed velocity $\text{Ready}$

Longitudinal net force $\text{Ready}$

Net work from launch $\text{Ready}$

Kinetic energy $\text{Ready}$

Potential; zero at rail origin $\text{Ready}$

Maximum energy-ledger residual $\text{Ready}$

Reached turns $\text{Ready}$

Available trajectory duration $\text{Ready}$

Acquired force stations $\text{Ready}$

Work from acquired force $\text{Ready}$

Acquired minus reference work $\text{Ready}$

DRAW / LAUNCH / MEASURE

Let signed area explain the turn

Cart mass

m

1.00\,kg

Initial position

x_0

0.50\,m

Initial signed velocity

v_0

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

Observation time window

T

8.00\,s

Force at 0 metres

F_{0}

3.00\,N

Force at 1 metres

F_{1}

3.00\,N

Force at 2 metres

F_{2}

1.00\,N

Force at 3 metres

F_{3}

-1.00\,N

Force at 4 metres

F_{4}

-3.00\,N

Force at 5 metres

F_{5}

-3.00\,N

Force at 6 metres

F_{6}

-3.00\,N

Replay fraction

\tau

0.35\,

Force scan intervals

n_s

24.00\,

Force noise bound

\eta_F

0.00\,N

A point-mass cart on a horizontal rail. The seven handles specify the complete longitudinal net force, linearly interpolated between metre marks. Weight and support cancel vertically; there is no drag or wall collision. Force is independent of velocity and time.

Time-domain motion uses analytic constant, harmonic or hyperbolic segments. Knot crossings and turns are located as events. A rest point with zero force remains in equilibrium; an unstable equilibrium may be approached without a finite-time turn. Leaving the modelled rail ends the replay with its exit velocity retained.

Potential is the negative exact integral of the declared force, with zero at the rail origin. A negative kinetic-energy value inferred from an unreachable position is not used to jump a barrier. Newtonian motion and the separate work integral form the ledger check.

Force acquisition is a spatial calibration scan across the entire rail, including places the cart may not reach. Positions are exact; force has bounded deterministic noise in newtons. The estimator integrates only the recorded piecewise-linear force, including partial intervals. It does not receive the force law or true energy.

A scan that misses force knots can bias work even without noise. Changing mass, launch state or replay preserves the force record. Editing the force or sensor clears it. The cart and supports are schematic; force and velocity glyphs have separate fixed gains. Scrub the gold clock; there is no autoplay.

Ideal teaching model and simulated force acquisition; explicit event and sampling boundaries.

Start with the essentials

Focus question: Does signed force area predict a turn?
One-sentence intuition: Independent motion and force integrals should agree within numerical error.

Core mathematical model

Signed area

W_{0\to x}=\int_{x_0}^{x}F(\xi)\,\mathrm d\xi

Work changes sign with displacement and force; returning is not accumulated absolute area.

Independent time-domain motion

\begin{aligned}m\ddot x&=F(x)\\F(x)&=F_i+a_i(x-x_i)\\x_i&=i\,\mathrm m\end{aligned}

Each segment has a constant, trigonometric or hyperbolic solution; turns and knot crossings restart it.

A separate ledger

\begin{aligned}K&=\tfrac12m v^2\\U(x)&=-\int_0^x F(\xi)\,\mathrm d\xi\\\Delta K-W&=0\end{aligned}

Potential is referenced to the rail origin. The ledger checks the time solution against an independently integrated force.

Only acquired force enters the estimate

\widehat W=\sum_i\frac{F_i^{\mathrm{obs}}+F_{i+1}^{\mathrm{obs}}}{2}\,(x_{i+1}-x_i)

The estimator interpolates measured force for partial intervals; it never receives true kinetic energy.

Common difficulties

Model and evidence

Typical misconceptionEvery zero-speed point is a stop, and a smooth data curve is exact.

Better mental modelInspect the force at rest and the event history; sparse force data can miss knots.

Run the experiment

01
Draw a force profile
Drag blue force knots; vary the initial velocity and mass.
What to observe: The same signed force drives the 3D cart and its work integral.
02
Find a real turn
Compare Harmonic turns, Barrier and return, and Initial reverse motion.
What to observe: Velocity passes through zero with a nonzero restoring force; work unwinds on return.
03
Separate rest from asymptotic approach
Compare Equilibrium with Asymptotic approach.
What to observe: Zero speed at zero force is equilibrium; the critical unstable state is approached without crossing.
04
Measure the area
Acquire with a scan that includes force knots, then use a sparse scan and add noise.
What to observe: The data estimator cannot recover force detail that was not sampled.

Core mathematical model

Signed area

Independent time-domain motion

A separate ledger

Only acquired force enters the estimate

Common difficulties

Model and evidence

Run the experiment

Draw a force profile

Find a real turn

Separate rest from asymptotic approach

Measure the area