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Calculating kinetic energy

Mathematical expressions, which quantify how the stored energy in a system depends on its configuration (e.g. relative positions of charged particles, compression of a spring) and how kinetic energy depends on mass and speed, allow the concept of conservation of energy to be used to predict and describe system behavior. Created by Sal Khan.

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Video transcript

- [Instructor] In this video, we're gonna talk about kinetic energy and we're also gonna 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. And 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 100 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 gonna give you the formula for kinetic energy, but then we are going to derive it. So the formula for kinetic energy is that it's equal to 1/2 times the mass of the object, times the magnitude of its velocity squared, or another way to think about it, its speed squared. And so 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 gonna be 1/2 times his mass, which is 100 kilograms, times the square of the speed, so times four meters squared per second squared, have to make sure that we square the units as well. And this is going to be equal to 1/2 times 100 is 50 times four is 200 and 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, or this is the units 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 gonna be 1/2 times the mass, 25 kilograms times the square of the speed here, so that's gonna be 16 meters squared per second squared. And then that gets us. We're essentially gonna have 1/2 times 16 is eight times 25, 200, and we get the exact same units and so we can go straight to 200 joules. So 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. And so their energy due to their motion, they have the same capacity to do work. Now, some of you are thinking, where does this formula come from? And 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. So 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 that 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. And so what is the force the same thing as? We know that the force is the same thing as mass times the acceleration. And we're going to assume that we have constant acceleration just so that we can simplify our derivation here. And then what's the distance that we're gonna travel. Well, the distance is gonna 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 gonna be the velocity divided by the acceleration. Think about it. If you're going, trying to get to a velocity of four meters per second, and you're accelerating at two meters per second, per second, four divided by two is gonna leave you with two seconds. And if you're starting at a speed of zero and you're going to a magnitude of a velocity or a speed of v, and you're assuming constant acceleration, your average velocity is just gonna be v over two. So this is just v over two. 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 two, mv squared over two, 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. And if you don't have energy loss it could in theory, do this much work.