---
title: "One-Dimensional Potential Landscape Interactive Lab | Sandbox Physics"
description: "Can you predict a particle's motion just from a graph of its potential energy? Most of it. Draw a line at the total energy: the particle can only be where the landscape lies below that line, and it turns around exactly where the two meet. The curvature at the bottom of a valley gives only the frequency of small oscillations. A valley as flat as a fourth-power curve has zero curvature there, so that frequency comes out zero, yet the valley still traps the particle."
canonical_url: "https://sandboxphysics.com/labs/potential-landscape/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/potential-landscape/"
last_modified: "2026-10-22"
---

M044 · Potential energy and turning points

# One-Dimensional Potential Landscape

**Can you predict a particle's motion just from a graph of its potential energy?** Most of it. Draw a line at the total energy: the particle can only be where the landscape lies below that line, and it turns around exactly where the two meet. The curvature at the bottom of a valley gives only the frequency of small oscillations. A valley as flat as a fourth-power curve has zero curvature there, so that frequency comes out zero, yet the valley still traps the particle.

Reshape the well by changing its curvature, quartic term and tilt. Drag the release point, or set position and velocity together in phase space, and compare the allowed region, the exact turning points and the real period with the small-oscillation prediction.

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

The model is an ideal point particle moving along a horizontal guide in a given polynomial potential, with all quantities in SI units. The 3D force probes and the bands marking the allowed region are display aids; the Lab does not simulate rolling, a physical barrier or an actuator mechanism. The vertical curve is an energy graph, not a real track. Gray contours are drawn at the release energy, and the recorded motion is teal. Playback recomputes the motion from saved checkpoints. The half-step comparison covers at most the first ten seconds; energy drift is checked at every integration step.

Physics tutorial

## The energy graph predicts a whole trajectory

**Background** A potential-energy graph gives the energy stored at each position. Its slope gives the force, its flat points are equilibria, and a horizontal line at the total energy marks every position a classical particle can reach. The curve is not a track: the particle moves along the horizontal axis, and the vertical axis is energy.

**Why it matters** OpenStax University Physics, section 8.4, uses potential-energy diagrams to find stable points and turning points. This Lab goes further: you can reshape the well, follow the motion in phase space, compute the period by integration and check how well the simulation conserves energy.

Start with the essentials

- **Focus question**: Which predictions still hold when the bottom of the well is not shaped like a parabola?
- **One-sentence intuition**: The turning points come from the full energy equation, but the small-oscillation frequency depends only on how sharply the well curves at its bottom.

### Core mathematical model

#### A potential you can reshape

$$
U(x)=\frac{kx^2}{2}+\frac{qx^4}{4}+fx
$$

The quadratic and quartic coefficients are never negative. If at least one of them is positive, the well traps the particle. The tilt term moves the equilibrium to one side.

#### Force and stability

$$
F(x)=-U\prime(x),\qquad U\prime(x_*)=0,\qquad \omega_0^2=\frac{U\prime\prime(x_*)}{m}
$$

If the curvature at the minimum is positive, small oscillations are simple harmonic. If the curvature is zero, the higher-order terms decide whether the minimum is stable.

#### Allowed positions and turning points

$$
E=\frac12mv^2+U(x),\qquad U(x)\le E,\qquad U(x_\pm)=E
$$

The speed is zero at a turning point. A particle released at rest at the equilibrium has an allowed region that shrinks to a single point.

#### Period at finite amplitude

$$
T(E)=\sqrt{2m}\int_{x_-}^{x_+}\frac{dx}{\sqrt{E-U(x)}}
$$

The Lab evaluates this integral separately from the simulation. A cosine substitution removes the simple blow-up at each turning point, and the readout shows how much the result changes at half the resolution.

#### Energy drift and the small-oscillation reference

$$
\delta E(t)=E(t)-E(0),\qquad x_{\mathrm{lin}}(t)=x_*+(x_0-x_*)\cos\omega_0t+\frac{v_0}{\omega_0}\sin\omega_0t
$$

The reference curve holds only near the equilibrium. When the small-oscillation frequency is zero, the reference is motion at constant velocity; that is the linear approximation, and it does not say the real minimum is neutrally stable.

### Common difficulties

#### Energy is not height

**Typical misconception** The particle slides downhill along the plotted curve.

**Better mental model** The vertical axis is energy. The particle actually moves back and forth along the horizontal axis.

#### Force comes from the slope

**Typical misconception** The force is largest where the potential energy is highest.

**Better mental model** The force depends on the slope at that point, not on the height. In a tilted well the equilibrium can even sit at a nonzero potential energy.

#### Zero curvature calls for higher-order terms

**Typical misconception** If the small-oscillation frequency is zero, the minimum must be unstable.

**Better mental model** A positive quartic term still traps the particle. It oscillates with a finite period that depends on the amplitude.

### Run the experiment

1. 01

   #### Check a harmonic well

   Select the Harmonic reference preset and press Acquire full record.

   **What to observe:** The period computed between the turning points matches the textbook harmonic period and the time between repeated turns in the trajectory.

2. 02

   #### Reshape the well

   Drag the orange stiffness handle, change Quartic confinement and compare the force with its linear prediction near the equilibrium.

   **What to observe:** The release energy and the turning points change together, and the controls show the actual coefficient values.

3. 03

   #### Flatten the minimum

   Select the Flat quartic minimum preset, then make the release displacement smaller.

   **What to observe:** The small-oscillation frequency stays at zero, but the actual period gets longer as the release gets smaller.

4. 04

   #### Move the equilibrium

   Select the Tilted landscape preset, then drag the release point in the phase-space plot in both directions.

   **What to observe:** The equilibrium moves. Changing the starting velocity changes the allowed interval, but the potential itself stays the same.

Physical conclusion

A flat quartic minimum is stable even though its small-oscillation frequency is zero.

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)