Energy is Conserved

54 Energy is Conserved

Now that we are comfortable with the idea of work we can go back to the work energy theorem and see how it gives rise to the idea of conservation of energy. Let’s see how this works out using a specific example: gravity.

Exercise 54.1: Gravitational Potential Energy

So far, we have modeled gravity as a constant downwards force, i.e. $\vec F = -mg \hat y$ .

A. Consider an object that moves from an initial position $\vec x_{\rm i}$ to a final position $\vec x_{\rm f}$ . Find the work done by gravity.

B. Suppose that the object from part A is only subject to the force of gravity. Then, the total work is just the work due to gravity. Plug in your expression for work into the work–energy theorem, and then collect everything that has an “f” subscript in the right hand side of the equation, and everything that has an “i” subscript in the left hand side of the equation. In English, what is this equation telling you?

We have found that the quantity $E\equiv K+mgy$ doesn’t change, i.e. $E$ is a conserved quantity. We call $U=mgh$ the gravitational potential energy, and $E=K+U$ is the total energy. With these definitions, our final result is simply $E_f=E_i$ , i.e. energy is conserved.

Important: We have found that gravitational potential energy is defined as $mgh$ , where $h$ is the “height.” Now, suppose that we define the origin of our coordinate system to be the ceiling, so that our “height” relative to the ceiling is actually negative. Since $h$ is negative, then our potential energy is negative. Is that an issue?

The answer is: “No.” We can define the origin wherever we want to. If the potential energy comes out negative, so what? The math tells us that $U=mgh$ , and so that’s what it is.

This is a big deal. As you know from momentum, conserved quantities are a wonderful cheat code: if we run into a “before and after” problem, and we don’t care about what happens in the middle, then conserved quantities are usually the best way to solve the problem. Let’s see how this plays out for gravity.

Exercise 54.2: Using Energy

You drop a ball from a height $h$ . How fast is the ball moving when it hits the ground?

We will solve this using conservation of energy. This is our first time using energy, so we will go through the solution slowly.

Step 0- What’s the story?

This is a “before and after” problem: it starts when we release the ball from height $h$ , and ends right as the ball hits the ground. Moreover, we don’t care about what happens in between, so we should try solving it using conservation laws. Momentum is obviously not conserved because the velocity is changing. Thus, we try energy. As a general rule of thumb

Before and after problems that involves trading height for speed should be solved using conservation of energy.

Step 1: Draw a picture. The picture should show both the before and the after. In addition, you should draw the forces acting on the ball! It will become clear why over the next two chapters.

Step 2 and 3: Find relations and solve. We already figured out this is a conservation of energy problem, so go ahead and use your picture to set up the conservation of energy equation. When you do so, make sure every force in your diagram is accounted for by a term in the energy equation. Then solve for the ball’s final speed.

We have found that the gravitational potential energy is given by $U=mgy$ . Our discussion helps us understand the origin of the name potential energy. When a mass is lifted a height $h$ , it has the potential to transform that height into velocity (kinetic energy) by falling. As the ball falls, the system is pouring energy from the potential energy cup into the kinetic energy cup.

Now- could you have solved this problem using the kinematic equations? Yes. But- you would have had to set up two equations, and solve for two unknowns. The energy approach is much simpler. This is why I argued in the chapter on momentum that you should

Always solve “before and after” problems using conservation laws.

Let’s drive this point home with another problem.

Exercise 54.3: Cannonball

A cannon stands atop an ocean cliff of height $h$ . It fires a cannonball with velocity $\vec v$ at an angle $\theta$ relative to the horizontal. How fast is the cannonball moving when it lands on the ocean?

This is a before/after problem, so you should solve this problem using conservation of energy. If you get stuck, use the slider to reveal my diagram, and take it from there.

This is pretty awesome: this was a 2D problem, but we were able to quickly solve it using only one equation. Hopefully you’re starting to see why conservation of energy is so powerful!

Key Takeaways

54 Energy is Conserved

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