Kinetic energy | Physics | Khan Academy

What's common between your morning hot coffee and a beautiful song coming from a guitar? To answer that question, we need to explore what kinetic energy is, and that's what we'll do in this video.

But let's zoom out a little bit. What exactly is energy to begin with? Well, I always found defining energy can be very hard; in fact, it can be very circular, so it could be hard to understand, actually. Instead of trying to define it over here, let's try to understand its features and see why it's so useful for us.

So, the first feature of energy is that it is a scalar quantity. Whenever I hear that something is scalar, I am always excited about it because scalar means it has no directions, it can be easily added up—it's just a number. So, that's awesome! It's a number that is measured in the standard unit of joules. You can have kilojoules, megajoules; you can also have non-standard units like calories that we use in our day-to-day life. But the standard unit of energy is joules.

So energy, so far, think of it as a number with no direction, which is measured in joules. Okay, what else? Well, another important thing is we can categorize energy into three broad categories. The first one is energy due to motion, which we call kinetic energy—the focus of this video. For example, a moving baseball has kinetic energy. If the baseball is moving faster, it has more kinetic energy.

Okay, energy assigned to moving things due to their motion. All right, what else? You can also assign energy based on where they are located, which we call potential energy. In fact, this is slightly more nuanced, and this will be the focus of the next video. But this is more interesting because potential energy is not just a number that you assign to a thing like kinetic energy; it's a number that you assign to a group of things based on their relative positions.

For example, if you consider the Earth and the ball as a group, then based on how far they are, you can assign a number to it—the potential energy. Now, if that distance changes, you'll get a different potential energy. So yes, there are nuances to it, but we'll tackle that in the next video, so don't worry too much about it.

Okay, what else? We have covered energy due to motion; we also have energy based on their relative positions. What else? Well, we can also have energy due to electromagnetic radiation. Think about light, X-rays, and all of those things—they too have energy, and that's something we'll tackle in a future unit altogether.

But now comes the big question: why should we care about energy in the first place? Here's why. Think about any process happening in nature. For example, nuclei fusing together in the core of our sun, or maybe leaves absorbing sunlight to create glucose—photosynthesis. Or consider a roller coaster that’s just speeding down. In all these things, a lot of changes are happening, but the one thing that doesn't change, turns out, is the total energy.

The total energy, it turns out, stays the same throughout the process. This means maybe kinetic energy would decrease, but that would result in an equal increase in, say, potential energy or maybe the energy of electromagnetic radiation. Whatever you do, the total energy never changes, and that's the reason we care about it. We call this the conservation of energy.

This is awesome because we can use this to predict stuff. For example, if I want to know how much the speed of the roller coaster is as it falls down, as it comes down a little bit, I can use energy conservation. I can use the fact that the total energy is conserved to make that prediction. We love doing this because energy is a scalar quantity; we can just directly add them up. We don't have to worry about their directions, and that's why sometimes using energy to predict things is much easier than using other things, like, say, forces or accelerations, because they are vector quantities.

So ultimately, that's the reason why we love energy—because it's a conserved quantity. Total energy throughout a process stays the same. All right, so with that in mind, for the rest of the video, let's just focus on kinetic energy.

So how do you calculate the kinetic energy of any moving object? Well, kinetic energy of any moving object is given by the expression: half mv squared, where m is the mass of the object and v is the speed of the object. Think of it as the magnitude of the velocity; I don't care about the direction, I only care about the speed squared.

So you can immediately see that kinetic energy is directly related to mass and the speed, and that makes sense because it's the energy of motion. So, more the speed, more the kinetic energy; but you also see, more the mass, more the kinetic energy. So, two objects can have the same speed but have different kinetic energies because they could have different masses. So mass also matters.

Now, one thing we can immediately see from this is that we can try to look at the units of energy by looking at this formula. If you look at the unit, you get mass, which is in kilograms, and speed, which is in m/s, but it's squared, so it's m²/s². But we know that this is energy, so that is the same thing as joules.

So we can now see what joules really is in the more fundamental units. Joules, at the end of the day, is basically kg m²/s². But let's try to get a feeling for joules by taking a few examples.

So let's take an example. Let's say we throw a baseball, and we know the mass of the baseball—let's say it's about 0.15 kg. And let's say this is a baseball throwing at about 90 mph—a fastball—so which roughly translates to about 40 m/s. What is the kinetic energy of this baseball? Why don't you pause the video and see if you can find this out yourself?

All right, let's do this. So kinetic energy is half mv². We know the mass; we know the speed, so we can just plug in. So that makes it half into 0.15 * 40 squared. So 40 squared, I can directly write as 1600. Four squared is 16, so you get 1600.

And yeah, we can simplify this. This makes it 8, so you get 800, and I can do this in my head—so 8 * 5 is 40; 8 * 1's are 8, so 12. So I get 120, and then these two zeros and then I have the two decimals, so one, two. So I get 120.

Let me just erase these zeros, so I get about 120 joules of kinetic energy. Remember, these are realistic numbers; this is basically a fastball, so you can kind of get a feeling for the amount of joules that we deal with in our daily life.

All right, let's take another example—not a daily life example. Let's take the example of a space probe that's going around Jupiter as we speak right now—it's happening. The mass of the space probe is given, and we know the kinetic energy of that space probe. In fact, kinetic energy keeps changing as it goes through the orbit, but at one particular moment, let's say the kinetic energy is this much.

We want to figure out what the speed of that space probe is. Again, great, I need you to pause the video and see if you can figure this out yourself first. All right, again, I know my kinetic energy is half mv², but this time you need to figure out what v is. K is given to me; m is given to me, so what I try to do next is I try to isolate v.

So let's rearrange this to isolate v. I'll multiply by 2 on both sides and divide by m on both sides, so I'll get 2K/m = v². But I don't want v²; I want v. How do I do that? Well, I take a square root on both sides.

So let me do that—take a square root on both sides. Now I'll have v on the right-hand side, so I'll just write—let me just write that over here—so v equals square root of (2K/m). Now we can plug in, so if we plug in, what do we get?

We have two times the kinetic energy given to me. I need to be a little bit more careful now—these are big numbers, so let me just write them down first and then divide by m, so m is given to me as 1.59 * 10^3. All right, so the first thing I do is I simplify this. I'll take my calculator and do that over here. So 2 * 4.33—I’ll ignore the 10 powers for now.

Okay, divided by 1.59. That gives me 5.4, if I round it—54, so let’s write that. So that gives me v equals—let me write that over here—square root of 5.45 times 10 to the power… what is this? Well, 10 to the power 12 / 10^3 is 10^9.

Now I can take a square root of this, but taking the square root of 10 to the 9 is difficult. Taking the square root of 10 to the even numbers is easy because it just becomes half; 10 squared, the square root becomes 10 to the 1. 10^4, the square root becomes 10², so I just have to take this.

Before taking the square root, let me convert this into an even number. One way to do that is basically—I can do this: 54—I’ll take 110 out, so I'll get 54.5 times, since I took one 10 out from here, I'm remaining with only 8 tens over here, so 10 to the 8. It’s an even number. Now I can take the square root. I know this will be 10^4 after the square root, and so I just have to take the square root of 54.5, which I'll do again over here—let me just take the square root.

Where's the square root? I can't find the square root. Okay, I just needed to do scientific. Okay, so now 54.5 – take a square root of that. I get 7.38. So I get about 7.38 times 10^4 m/s.

Now, remember, in general, when you take a square root of any number mathematically, the answer is plus or minus. You have two answers for that because square of 2 is 4, square of -2 is also 4, right? So technically I should get a positive and a negative number, but since v is the speed, speeds can't be negative—that's why we know we only take a positive number, and so our answer is a positive number.

That brings us to our original question: what's common between these two? Well, a hot coffee, we say, has thermal energy in it, right? The hotter it is, the more thermal energy. But if you were to zoom in when things are hot, basically their atoms are jiggling faster, or they're moving very fast—in other words, they have more kinetic energy.

So thermal energy, at a microscopic level, is basically kinetic energy. Similarly, if you look at sound energy, we might think of it as a different kind of energy, but again, if you were to zoom in and look at it, what exactly is sound energy? Sound energy is where atoms or molecules of the medium, like, say, the air, are basically moving back and forth in unison.

We call that a sound wave, but at the end of the day, that energy is basically the energy of motion of these atoms and molecules. So again, it is kinetic energy at a microscopic level. So what's common between them is they're not new forms of energy; they are both kinetic energies at a microscopic level.

Now, of course, remember, because there are also multiple particles and they're all interacting with each other based on their relative positions, there's also potential energy. So yes, there's both kinetic and potential energy. But the takeaway is it's not a new form of energy. Thermal energy and sound energy are basically kinetic and potential energies at the micro level.