---
title: "Motion Graph Studio Interactive Lab | Sandbox Physics"
description: "Can one drawn curve determine the whole motion? A position curve does: its slope is the velocity, and the way the slope changes is the acceleration."
canonical_url: "https://sandboxphysics.com/labs/motion-graph-studio/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/motion-graph-studio/"
last_modified: "2026-10-28"
---

M001 · Kinematics

# Motion Graph Studio

**Can one drawn curve determine the whole motion?** A position curve does: its slope is the velocity, and the way the slope changes is the acceleration. A velocity curve only tells you how far the object moves, so you also need its starting position; an acceleration curve needs the starting velocity too.

Draw the position, velocity or acceleration curve and watch a 3D cart follow it. Slopes and areas, with their signs, show when the cart turns back and why distance and displacement differ.

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

The motion is prescribed for teaching. The Lab states plainly how the curves are interpolated and what the measurements can and cannot show.

Physics tutorial

## Motion Graph Studio

**Background** A motion graph shows two things: its slope tells how fast a quantity changes, and the signed area under it tells how much has built up.

**Why it matters** Draw one of the three curves and give the starting values, and slopes and areas fix the other two. Watching all three beside a moving 3D cart makes those links concrete.

Start with the essentials

- **Focus question**: Can one drawn curve determine the whole motion?
- **One-sentence intuition**: Integrating a graph needs a starting value, and area below the axis counts as negative.

### Core mathematical model

#### Velocity and acceleration are slopes

$$
v=\frac{\mathrm dx}{\mathrm dt},\qquad a=\frac{\mathrm dv}{\mathrm dt}
$$

The three curves describe one motion, so editing the driver changes all three together.

#### Integrals need initial values

$$
\begin{aligned}v(t)&=v_0+\int_0^t a(s)\,\mathrm ds\\x(t)&=x_0+\int_0^t v(s)\,\mathrm ds\end{aligned}
$$

After a reversal, positive and negative areas can cancel.

#### Distance counts the trip back

$$
D(t)=\int_0^t|v(s)|\,\mathrm ds\geq|x(t)-x(0)|
$$

The zeros of the velocity split each polynomial piece into segments of one-way travel.

#### Natural boundary of the chosen driver

$$
q\in\{x,v,a\},\qquad q^{\prime\prime}(0)=q^{\prime\prime}(T)=0
$$

The input spline has a continuous second derivative. A natural boundary does not mean zero speed.

### Common difficulties

#### What a natural spline boundary means

**Typical misconception** A natural spline boundary means the cart is at rest at the ends.

**Better mental model** You choose position, velocity or acceleration as the driver, and seven equally spaced knots define a natural cubic spline for it. Only the driver’s second derivative is zero at the two outer ends; the cart does not have to start or end at rest. The spline can also overshoot the knot values.

### Run the experiment

1. 01

   #### Select the driver

   Compare Draw position, Uniform motion and Constant acceleration.

   **What to observe:** Initial values matter only for curves obtained by integration.

2. 02

   #### Edit and reverse

   Drag a blue knot across zero, then drag the gold clock.

   **What to observe:** While the velocity is negative, the cart moves back along the rail.

3. 03

   #### Separate distance and displacement

   Choose Reverse and return and drag the clock to the end.

   **What to observe:** The displacement can return to zero even though the cart has traveled some distance.

4. 04

   #### Check the integral

   Compare exact position with the separate trapezoid trace.

   **What to observe:** The sampling error of the trapezoid sum stays visible; it is not forced to zero.

Physical conclusion

Distance tells you how far the cart traveled; displacement tells you where it ended up relative to the start.

## Related experiments and articles

Interactive Physics Textbook · Reading path

### Motion, Force and Energy

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How do we describe a motion?**](https://sandboxphysics.com/learn/mechanics/describing-motion/)

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)