Work-Energy Principle Example | Energy and Momentum | AP Physics 1 | Khan Academy

So the work energy principle states that the net work done on an object is going to equal the change in kinetic energy of that object. And this works for systems as well. So, the net work done on a system of objects is going to equal the change in the total kinetic energy of the objects in that system.

Now that sounds really complicated and technical, but I like to think about the work energy principle as a shortcut. This is a really nice shortcut that lets me determine the change in kinetic energy without having to do a bunch of complicated conservation of energy equations or kinematic formulas. The catch is that I need to know how to figure out what the net work is.

So how do you figure out net work? Well, the formula for work done is ( W = Fd \cos(\theta) ). Since this formula for the work energy principle relies on net work, this has to be the magnitude of the net force times the magnitude of the distance traveled times cosine of theta. Remember, this theta has to be the angle between—not any angle—between the net force direction and the direction of motion.

And so let's try this out. How do you use this thing? Let's kick the tires. Let's say there's a satellite moving to the right and there's a net force on this satellite. Now this net force could go in any direction. If the net force has a component in the direction of motion, then the net work's going to be positive.

If so, anything here from like negative 90—well, I guess like negative 89.9—because 90 would be perpendicular. From any force, like 89.9 negative to positive 89.9, you've got a component in the direction of motion. That means you're going to be doing positive net work, and that means the change in kinetic energy will be positive because it just equals that number. That means kinetic energy increases; you're going to be speeding up.

And that kind of makes sense intuitively. If your force is in the direction of motion, you're speeding up. What about the other case? What if your net force points in the opposite direction of motion? Well now the net work's going to be negative. You'll have a negative change in kinetic energy. In other words, you're going to slow down.

And if the net force points perpendicular, well then you're not doing any work because cosine of 90 is going to be zero. No net work would be done; there's going to be no change in kinetic energy. That doesn't mean you stop; it just means you're not going to speed up or slow down. This does something. You might be like, "Doesn't it do something?" Yeah, you're going to drift upward; you're going to start changing your direction. But this is not going to be doing any work on you at that moment.

And so just to be clear, let me—let's just try a complicated one here. Let's say this force goes in some direction. Let's say your velocity even goes down. So, I mean, maybe your satellite is heading downward and your force is going to go in any direction. Well, if it goes this way exactly backwards, it can be 180; you're going to be doing negative work. You're going to be slowing down, decreasing kinetic energy, and you're not gonna change direction.

If you're like this, you have a component opposite, so you're going to be slowing down and changing direction. This will just be changing direction; you're not speeding up or slowing down at that moment. This will speed you up and change your direction, and finally, this will just be speeding you up and you will not be changing your direction.

So the work energy principle is convenient to just get a conceptual or qualitative idea of what's going on, and it can obviously also give you an idea of how to calculate things. So let's try one—we actually have to get a number.

So, let's say there's a hot air balloon, and it's a 300-kilogram hot air balloon drifting to the left. It had an initial speed of seven meters per second, and it’s traveling a total of 50 meters to the left during this journey. Now there are going to be forces on this hot air balloon; obviously, there are going to be gravity and some buoyant force.

But because these are perpendicular to the direction of motion, they do no work. So, when we're going to use this work energy principle, they're not even going to factor in. We don't even have to know these since they were perpendicular and did no work. You only consider the forces in the direction of motion.

So, say there was a wind gust helping you to the left here of 200 newtons, but this is a big bulky balloon—not that aerodynamic—and so there was a drag force from air resistance of 104 newtons to the right. What we want to know is we want to determine the final speed of the hot air balloon after it travels 50 meters directly to the left with the forces shown.

Now there's lots of different ways to do this. You know, Newton's laws; you can do kinematic formulas, all kinds of stuff— even momentum, technically impulse. But the easiest—I'm pretty sure the easiest way to do this is just going to be the work energy principle, which states that the net work done is going to equal the change in kinetic energy.

So let's go ahead and do it. So, we know that net work is equal to the magnitude of the net force times the magnitude of the distance traveled times cosine of the angle between them. What's changing kinetic energy mean? Well, change in anything is final minus initial.

So since kinetic energy is (\frac{1}{2} mv^2), this is just going to be (\frac{1}{2} m v_{\text{final}}^2 - \frac{1}{2} m v_{\text{initial}}^2). So, this is going to be the change in kinetic energy: final minus initial.

All right, let's plug in numbers here. So, net force—how do we get that? Vertical pieces don't matter; we're just looking horizontally here. Those are the only ones that are going to affect it. These vertical ones just cancel. So we have 200 to the left, 104 to the right.

So we have to subtract those to get 96 to the left, and we just want magnitude. So I'm going to get 96 newtons to the left—not negative or anything; I'm just taking magnitude—times the distance traveled. We know that's 50, so we get (96 \times 50, \text{meters} \cdot \cos(\theta)).

Now this is careful; you might be like, "Oh, 180 here." But no, the net force points to the left. This 200 is winning here, so leftward net force and the leftward direction—the angle between these two is zero. Cosine of zero is just one; we’re maxed out here.

So, net force points in the direction of motion. So, equals—let's plug in the rest of this: (\frac{1}{2}) the mass is 300 kilograms times (v_{\text{final}}^2) (that’s what we want to determine) minus (\frac{1}{2} \cdot 300, \text{kg} \cdot v_{\text{initial}} = 7, \text{m/s}). So we got 7 meters per second squared.

Well, (96 \times 50) is going to come out to (4800); that means that’s the net work done. Notice that’s joules—that’s how much energy we’ve added. That’s the change in kinetic energy. So we know the net work is changing kinetic energy; we’ve added (4800, \text{joules}) of kinetic energy.

That’s going to have to equal (\frac{1}{2} \cdot 300) is 150 kilograms times (v_{\text{final}}^2) minus if you take (\frac{1}{2} \cdot 300 \cdot 7^2) you’re going to get (7350). So this is how much energy the hot air balloon started with initially.

So I have to move that to the left. We add those together; I'm going to get (12150, \text{joules}) is how much kinetic energy the balloon ends with, and that’s got to equal (150, \text{kg} \cdot v_{\text{final}}^2).

I could divide (12150) by (150), and you get exactly (81). And that’s going to equal (v_{\text{final}}^2). And if you take a square root of that, you get exactly (9). So the final velocity of this hot air balloon is (9, \text{m/s}).

It’s sped up; that’s not surprising. This net force was directed leftward, and the object was moving leftward. So we're doing positive work; we’re increasing the kinetic energy. We started with (7, \text{m/s}); we ended with (9, \text{m/s}).

And this is an example of how you use the work energy principle. So, to recap: the work energy principle states the net work is equal to the change in kinetic energy. This can help you just conceptually or qualitatively determine whether something's going to speed up, slow down, or change direction—or both.

And then quantitatively, you can use this to specifically solve for the change in kinetic energy as well as the final or initial speed something might have had.