---
title: "Work–Energy Graph Studio Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/work-energy-studio/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/work-energy-studio/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

Physics tutorial

## Work–Energy Graph Studio

**Background** One force profile gives two things: the motion over time and the signed area under the force-position curve, which is the work.

**Why it matters** Draw the force along a rail, follow the exact motion piece by piece through each turn, and then check recorded force samples against the separately computed energy.

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 misconception** Every point where the speed is zero is a stop, and a smooth curve through the data is exact.

**Better mental model** Look 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Work has a sign, and knowing that the speed is zero does not tell you what happens next.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)