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## High school physics - NGSS

# 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.

## Want to join the conversation?

- How is the average velocity equal to v over 2?(5 votes)
- This only holds when you are accelerating from 0m/s. Think about a velocity time graph. Constant acceleration from 0m/s to the final velocity would result in a straight line from 0m/s to the final velocity in m/s. It is intuitive to see that the average velocity in this time period would be the middle point of the line. This middle point would be the total velocity divided by 2. Therefore, average velocity is equal to final velocity divided by 2 when the acceleration is constant and the initial velocity is 0.(5 votes)

- What is the difference between joules and newtons?(1 vote)
- it blockes the sun because the moon is closer and the sun is futher so it makes it look smaller(0 votes)
- ha. ha. very funny.(1 vote)

## 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.