Calculating kinetic energy | Modeling energy | High school physics | Khan Academy

In this video, we're going to talk about kinetic energy, and we're also going to think about how to calculate it. So, you can already imagine, based on the word "kinetic," which is referring to motion, that this is the energy that an object has by virtue of its motion. When we talk about energy, we're talking about its capacity to do work.

So, just based on that early definition of kinetic energy, which of these two running backs do you think has more kinetic energy? This gentleman on the left, whose mass is a hundred kilograms and who is traveling at a speed of two meters per second, or the gentleman on the right, who has a mass of 25 kilograms and who's traveling with a speed of four meters per second? Pause this video and think about that.

All right, now let's think about this together. So, I'm first just going to give you the formula for kinetic energy, but then we are going to derive it. The formula for kinetic energy is that it's equal to one-half times the mass of the object times the magnitude of its velocity squared, or another way to think about it is its speed squared. Given this formula, pause the video and see if you can calculate the kinetic energy for each of these running backs.

All right, let's calculate the kinetic energy for this guy on the left. It's going to be one-half times his mass, which is 100 kilograms, times the square of the speed. So, times 4 meters squared per second squared. We have to make sure that we square the units as well. This is going to be equal to one-half times 100, which is 50, times 4, which is 200.

Then, the units are kilogram meter squared per second squared, and you might already recognize that this is the same thing as kilogram meter per second squared times meters, or these are really the units of force times distance. This is the unit of energy, which we can write as 200 joules.

Now let's do the same thing for this running back that has less mass. Kinetic energy here is going to be one-half times the mass, 25 kilograms, times the square of the speed here. So, that's going to be 16 meter squared per second squared. Then that gets us, we're essentially going to have one-half times 16, which is 8, times 25, which is 200.

We get the exact same units, and so we can go straight to 200 joules. It turns out that they have the exact same kinetic energy, even though the gentleman on the right has one-fourth the mass and only twice the speed. We see that we square the speed right over here, so that makes a huge difference. Their energy, due to their motion, shows they have the same capacity to do work.

Now, some of you are thinking, where does this formula come from? One way to think about work and energy is that you can use work to transfer energy to a system or to an object somehow, and then that energy is that object's capacity to do work again.

Let's imagine some object that has a mass m and the magnitude of its velocity or its speed is v. So, what would be the work necessary to bring that object, which has mass m, to a speed of v, assuming it's starting at a standstill? Well, let's think about it a little bit. Work is equal to the magnitude of force in a certain direction times the magnitude of the displacement in that direction, which we could write like that. Sometimes they use s for the magnitude of displacement as well.

So, what is the force? The same thing as we know that the force is the same thing as mass times the acceleration. We're going to assume that we have constant acceleration, just so we can simplify our derivation here. Then, what's the distance that we're going to travel? Well, the distance is going to be the average magnitude of the velocity, or we could say the average speed.

So, I'll write it like this, times the time that it takes to accelerate the object to a velocity of v. Well, how long does it take to accelerate an object to a velocity of v if you're accelerating it at a? Well, this is just going to be the velocity divided by the acceleration. Think about it; if you're trying to get to a velocity of 4 meters per second and you're accelerating at 2 meters per second squared, 4 divided by 2 is going to leave you with 2 seconds.

If you're starting at a speed of 0 and you're going to a magnitude of a velocity or speed of v and you're assuming constant acceleration, your average velocity is just going to be v over 2. So, this is just v over 2, and then we get a little bit of a drum roll right over here. We see that acceleration cancels with acceleration, and we are left with mass times v squared over 2, or mv squared over 2, which is exactly what we had right over here.

So, the work necessary to accelerate an object of mass m from zero speed to a speed of v is exactly this, and that's how much energy is then stored in that object by virtue of its motion. If you don't have energy loss, it could, in theory, do this much work.