Interpreting mechanical energy graphs

When we have a moving object for which mechanical energy can be treated as conserved (resistive forces can be ignored), plotting a graph of the potential energy can be a very useful tool for telling the story of how the motion takes place in time. In this case, the work-energy theorem simply becomes that as the system moves, there is no change in the kinetic plus potential energy. Or we can say that the energy at an initial time is equal to the energy at a final time:

$$\Delta \bigg({1 \over 2} \;  m v^2  + U\bigg) = 0$$

$${1 \over 2} \;  m v_i^2  + U_i = {1 \over 2} \;  m v_f^2  + U_f$$

A particularly useful way of writing this is to take the value of the total energy at the initial time — call it $E_0$ — and use the "initial energy = final energy" equation to write that the energy at any time is equal to $E_0$.

$$E_0 = {1 \over 2} \;  m v^2  + U$$

The potential energy, $U$, is a function of position and the velocity, $v$, is a function of time. This equation then tells us how the kinetic energy (and therefore the speed) changes as the object's position changes. This is a deep and valuable feature:

The conservation of energy relates an object's position (through the PE) to its speed (through the KE), allowing us to easily determine an object's speed when it reaches different places.

What the conservation law has done is to remove a time. We may be able to figure out how fast the object is going when gets to a certain place, but we don't know when it will get there.*

The potential energy can be any combination of the potential energies we have looked at — or could be any potential energy that arises from a more complex (but conservative) force.  Let's see how the story of the physical motion is coded in a graph by considering three specific cases.

Case 1: An open motion - Two positive charges

Here's an example. Consider a light, movable positive charge ($q$) approaching another positive charge ($Q$), that one heavy and fixed in space. In this case, the total energy will be the KE of the first charge plus the PE of the interaction. We know that the PE is $U_{elec} = k_c qQ/r$.  The graph of the PE as a function of the separation of the two charges is shown below (in unspecified units).

Let's use the graph to tell the story of the motion. Suppose that charge q approaches charge $Q$ starting from very far away with KE equal to $E_0$ (as shown by the yellow double-headed arrow).  As it gets closer (as $r$ gets smaller), the PE grows as shown by the green double-headed arrows. When charge $q$ reaches $r$ = 4, the total energy is still $E_0$, but some of it is now PE so the KE is reduced to KE₄ (shown as a red double-headed arrow). When the object reaches $r$ = 3, the total energy is still $E_0$, but more of it is PE so the KE is reduced further, to KE₃, as shown. As it goes in, towards the origin, the KE continues to drop as the PE rises, until, when the charge reaches $r$ = 1/2, the energy is now all potential. The KE is 0, so the charge stops. It cannot get to a distance closer than 1/2 since it doesn't have enough energy.

The key in reading a PE graph is to see that the KE is the gap between the total energy and the PE curve. Since KE is ½mv², it can never be negative. 

After the charge stops momentarily, it will now "roll back down the hill", going through the increasing values of KE as the PE decreases. Since KE is a scalar and doesn't care about direction, at any point where we determine the object's KE, it could be going in either direction. The graph only tells us about what the value of the KE is at any position.

Case 2: A bounded motion - Mass on a spring

As a second example, let's consider a cart on a horizontal spring. If we choose our coordinate so the the cart is at position $x$ = 0 when the spring is at its unstretched length, the PE of the spring is just ½kx². The graph looks like the one shown below.

For this case, let's start our cart connected to the spring at its rest length but moving to the right with a KE of $E_0$.

The point on the graph representing this instant is at $x$ = 0. The green arrow shows that KE is the full energy. If the cart is now moving to positive values of $x$, the spring pulls on it, doing work and slowing it down, at the same time increasing the PE stored in the spring. As we follow the cart to larger and larger values of $x$, the KE keeps decreasing until, when $x$ = 2.9, the KE is 0 — the PE is equal to the total energy — so the cart stops. There is a force on it so it begins to move back and runs "down hill" through decreasing values of PE and increasing values of KE (green arrows). As it runs to the left — to negative values of x, where the spring is compressed — the KE is reduced again until it is brought to 0 at $x$ = -2.9 and the cart again comes to a stop. The pattern of exchanging PE and KE keeps repeating as the cart oscillates back and forth.  Since we have assumed there are no resistive forces, it will keep oscillating forever (or, rather, for as long as that model continues to be reasonable).

Case 3: A complex PE - Molecular interactions

Suppose that there were a more complex potential energy, $U$, such as the one shown in the figure at the right. What would the motion of an object in this PE look like if it has an energy $E_0$? $E_1$?

This situation will become relevant when we look at the binding of atoms into molecules and chemical reactions

* If you wanted to find out an object's speed when it gets to a certain place using Newton's 2nd law, you would have to add up the changes in position by doing an integral over time. This is why when energy conservation was invented, it was called the first integral of Newton's law.

Joe Redish 11/5/11