M041 · Work and energy

Work–Energy Graph Studio

Can the area under a force-position graph predict how an object moves? Yes, as long as you also read the force where the speed runs out. The signed area under the force-position curve is the change in kinetic energy, so it gives the speed at every point and shows where it falls to zero. If the force there is not zero, the object turns back. If the force is zero there as well, the object can creep toward that point without ever reaching it.

Draw the force along a rail and follow the exact motion, segment by segment, through each turn. Then integrate recorded force samples and compare the result with the separately computed energy change.

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 change $0\,\mathrm{J}$

Kinetic energy $\text{Ready}$

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

Maximum work-energy 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 a force and measure its work

Use signed area to 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

The cart is a point mass on a horizontal rail. The seven handles set the full net force along the rail, which varies linearly between the metre marks. Weight and support cancel vertically, and there is no drag or collision with a wall. The force does not depend on velocity or time.

The motion in time is built from exact segments with constant, harmonic or hyperbolic form. Knot crossings and turnarounds are located as events. A cart at rest where the force is zero stays in equilibrium, and a cart can creep toward an unstable equilibrium without turning back in finite time. Leaving the modelled rail ends the replay, and the exit velocity is kept.

The potential energy is minus the exact integral of the given force, with zero at the rail origin. At a position the cart cannot reach, the energy balance would give a negative kinetic energy; that value is never used to carry the cart over a barrier. The Newtonian motion and the separately computed work integral are compared as a check.

A force acquisition scans the whole rail, including places the cart may never reach. Positions are exact, and the force carries a bounded, repeatable noise measured in newtons. The estimator integrates only the recorded force, joined point to point by straight lines, including partial intervals. It never sees the force law or the true energy.

If the scan misses the knots where the force curve bends, the work can come out biased even with no noise. Changing the mass, launch state or replay keeps the force record; editing the force or sensor clears it. The cart and supports are schematic, and the force and velocity arrows each have their own fixed scale. The finite record plays in physical time; drag the cart or gold clock to pause and replay.

This is an ideal teaching model with simulated force measurements. The page states exactly how turning events and sampling are handled.

Start with the essentials

Focus question: Does signed force area predict a turn?
One-sentence intuition: The motion solved over time and the work found by integrating the force are computed separately, and they should agree within numerical error.

Core mathematical model

Work as signed area under the force

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

Work changes sign with the direction of the displacement and of the force. On the way back the area is subtracted; it does not pile up as absolute area.

Motion solved in time

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

Within each segment the motion has a constant-acceleration, trigonometric or hyperbolic solution. The solution restarts at every turn and every knot crossing.

A separate energy check

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

Potential energy is zero at the rail origin. This check compares the time solution with a separate integral of the force.

Work estimated from recorded force only

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

For partial intervals the estimator interpolates the measured force. It never sees the true kinetic energy.

Common difficulties

Zero speed is not always a stop

Typical misconceptionEvery point where the speed is zero is a stop, and a smooth curve through the data is exact.

Better mental modelLook at the force where the cart is at rest and at the event history. Sparse force data can miss the knots of the force profile.

Run the experiment

01
Draw a force profile
Drag the blue force knots, and vary the initial velocity and the mass.
What to observe: The same signed force drives the 3D cart and sets its work integral.
02
Find a real turn
Compare Harmonic turns, Barrier and return, and Initial reverse motion.
What to observe: The velocity passes through zero while the restoring force is not zero, and the work is undone on the way back.
03
Tell rest from a slow approach
Compare Equilibrium with Asymptotic approach.
What to observe: Zero speed with zero force is equilibrium. The cart creeps toward the critical unstable state without ever crossing it.
04
Measure the area
Acquire with a scan that includes the force knots, then try a sparse scan and add noise.
What to observe: The estimate from the data cannot recover force details that were never sampled.

Core mathematical model

Work as signed area under the force

Motion solved in time

A separate energy check

Work estimated from recorded force only

Common difficulties

Zero speed is not always a stop

Run the experiment

Draw a force profile

Find a real turn

Tell rest from a slow approach

Measure the area