2.3.3.3. Potential-Energy Curves

Table of Contents

Reading Motion from Potential-Energy Curves

A potential-energy curve is a graph of potential energy $U$ versus position, usually one coordinate $x$. It is a very powerful way to understand motion without solving the full equations step by step. By looking at the shape of the curve, you can tell where an object can move, where it cannot move, where it speeds up, where it slows down, and where equilibrium occurs.

This chapter focuses on how to read information directly from the graph. The ideas rely on potential energy and conservation of mechanical energy, which are treated elsewhere. Here we use those ideas specifically for curves.

The Basic Graph

On a potential-energy curve, the horizontal axis is position and the vertical axis is potential energy. A particle moving under a conservative force has some total mechanical energy $E$. On the graph, this total energy is often shown as a horizontal line.

At each position, the kinetic energy is

$$
K = E - U(x)
$$

So the graph immediately tells you whether motion is possible. Since kinetic energy cannot be negative, the particle can exist only where

$$
E \ge U(x)
$$

If at some point $U(x) > E$, then $K$ would be negative, which is impossible in classical mechanics. That region is forbidden.

For motion on a potential-energy curve, the allowed region is where $E \ge U(x)$.
The kinetic energy is
$$
K = E - U(x)
$$
If $E = U(x)$, then $K = 0$.

Allowed and Forbidden Regions

Suppose a horizontal energy line cuts the curve in certain places. Wherever the energy line lies above the potential curve, the particle can move. Wherever the energy line lies below the curve, the particle cannot go.

These crossings are especially important. At a crossing point,

$$
E = U(x)
$$

so the kinetic energy is zero there. The particle momentarily stops and reverses direction. Such points are called turning points.

A particle trapped between two turning points moves back and forth in that region. A particle with enough energy to pass over a hill in the curve can continue into another region.

Rendering scientific sketch…

Allowed and forbidden regions on a potential-energy curve

Speed from the Curve

The potential-energy curve also tells you how the speed changes. Since kinetic energy is related to speed by

$$
K = \frac{1}{2}mv^2
$$

and also by

$$
K = E - U(x)
$$

we get

$$
\frac{1}{2}mv^2 = E - U(x)
$$

$$
v = \sqrt{\frac{2}{m}\left(E - U(x)\right)}
$$

This means the particle moves fastest where $U(x)$ is smallest, because then $E - U(x)$ is largest. It moves slowest near turning points, where $U(x)$ approaches $E$ and the speed approaches zero.

A particle is faster where the potential energy is lower, provided the total energy is fixed.
$$
v = \sqrt{\frac{2}{m}\left(E - U(x)\right)}
$$

Force from the Slope

The slope of the potential-energy curve tells you the force. In one dimension,

$$
F(x) = -\frac{dU}{dx}
$$

This is one of the most important ways to extract physics from the graph.

If the curve slopes upward as $x$ increases, then $\frac{dU}{dx} > 0$, so $F < 0$. The force points to the left.

If the curve slopes downward as $x$ increases, then $\frac{dU}{dx} < 0$, so $F > 0$. The force points to the right.

If the curve is flat, then $\frac{dU}{dx} = 0$, so the force is zero.

The steeper the curve, the larger the magnitude of the force.

The force is the negative slope of the potential-energy curve:
$$
F(x) = -\frac{dU}{dx}
$$
Positive slope means force to the left.
Negative slope means force to the right.
Zero slope means zero force.

Equilibrium Points

Equilibrium occurs where the force is zero, so on the graph this happens where the slope is zero. These are points where the curve has a horizontal tangent.

There are different kinds of equilibrium, and the curve shape reveals them.

If the equilibrium point is at a local minimum of $U(x)$, then a small displacement raises the potential energy, and the force tends to pull the particle back. This is stable equilibrium.

If the equilibrium point is at a local maximum of $U(x)$, then a small displacement lowers the potential energy, and the particle moves farther away. This is unstable equilibrium.

If the curve is flat over a region, then the force is zero there and the equilibrium is neutral.

Rendering scientific sketch…

Stable and unstable equilibrium

Potential Wells and Barriers

A low region in the curve is called a potential well. If the total energy line lies below the surrounding hills, the particle is trapped in the well and moves back and forth between turning points.

A high region in the curve acts like a barrier. To cross it, the particle must have total energy greater than or equal to the barrier height.

This idea is useful in many areas of physics. In classical mechanics, if the energy is less than the barrier, the particle cannot cross. In quantum physics, the situation becomes more subtle, but that belongs to a later part of the course.

Small Oscillations Near a Minimum

Near a stable equilibrium point, the potential-energy curve often looks approximately like a parabola. If the minimum is at $x = x_0$, then near that point the potential can often be written approximately as

$$
U(x) \approx U(x_0) + \frac{1}{2}k(x - x_0)^2
$$

This is the same shape as the potential energy of a spring. That is why many systems near stable equilibrium behave approximately like simple harmonic motion for small displacements.

The exact study of oscillations is covered elsewhere, but the key graph idea is simple. A smooth minimum often means restoring force and oscillatory motion nearby.

How to Analyze a Potential-Energy Curve

When you are given a graph of $U(x)$, a good method is to ask a few standard questions. First, where is the particle allowed to be for a given total energy $E$? Second, where are the turning points? Third, what is the direction of the force in each region from the slope? Fourth, where are the equilibrium points and are they stable or unstable? Fifth, where is the particle fastest and slowest?

The graph then becomes a complete visual summary of the motion.

Example of Interpretation

Imagine a particle with total energy $E$ inside a bowl-shaped potential curve. The energy line intersects the curve at two positions. The particle can move only between those two positions. At either edge, the kinetic energy is zero, so those are turning points. At the center of the bowl, the potential energy is minimum, so the kinetic energy is maximum and the particle moves fastest there. The slope on the right side is positive, so the force points left. The slope on the left side is negative, so the force points right. Both forces push the particle back toward the minimum, showing stable equilibrium.

Summary Table

Feature on graph	Meaning
$E > U(x)$	Motion allowed, kinetic energy positive
$E = U(x)$	Turning point, kinetic energy zero
$E < U(x)$	Forbidden region
$\frac{dU}{dx} > 0$	Force is negative, points left
$\frac{dU}{dx} < 0$	Force is positive, points right
$\frac{dU}{dx} = 0$	Force is zero, possible equilibrium
Local minimum of $U$	Stable equilibrium
Local maximum of $U$	Unstable equilibrium

Essential Rules

For one-dimensional motion in a conservative force field, potential-energy curves can be read using these core rules:
$$
K = E - U(x)
$$
$$
F(x) = -\frac{dU}{dx}
$$
Allowed motion requires
$$
E \ge U(x)
$$
Turning points occur where
$$
E = U(x)
$$
Stable equilibrium occurs at a local minimum of $U(x)$.
Unstable equilibrium occurs at a local maximum of $U(x)$.

Final Perspective

Potential-energy curves turn motion into a picture. Instead of tracking every instant of motion directly, you can read the force from the slope, the speed from the vertical difference $E - U(x)$, and the possible path from the allowed region. This makes them one of the most useful visual tools in classical mechanics.

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