Where does the work go or what happens to it?

We'll see that there are several different ways of converting work.

Let's start with the example of a force applied over a distance

to accelerate over to you but you'll need

to remember some results from weeks two and four.

Hey, well done.

That's a really important result so we'll revisit it here.

Mass m, initially at x0 with velocity v0, constant total force F in the x direction

accelerates it at a equals F over m.

Constant force gives constant acceleration.

So we can use equation three from kinematics

to relate the displacement x minus x0 to the initial and final velocities.

The total force is constant and parallel to the displacement so the work is

F times x minus x0.

Substituting for Newton's second law, we have work equals ma times x minus x0."

But remembering equation three,

let's multiply by one and write the one as a half times two.

Now we can substitute for 2a x minus x0.

That's just v squared minus v0 squared.

So now we have total work done equals a half mv squared minus a half mv0 squared.

This is the change in the quantity half mv squared.

That is such an important result that we give that quantity a name.

We call that quantity the kinetic energy.

Kinetic energy is defined as a half mv squared.

What we've just proved is that the total work done by all the forces acting on an object

is equal to the change in its kinetic energy.

Actually we've only shown this for constant force in one dimension

but it's true for variable forces as well and it's called The Work Energy Theorem.

Again let's check the units.

Mv squared is kilograms times meters per second squared.

Take one of the meters outside and

use Newton's second law to remind us that kilogram meters per second squared

is force units newtons and newtons times meters equals joules.

Let's do a question to get used to the units.

Just by the way there is a famous complication to this theorem

that was analyzed by Einstein in 1905.

We shan't go there now but we'll provide a link if you want to see

where that very famous equation comes from and why it is consistent with our definition

if v is very much less than the speed of light.

You may be wondering why does kinetic energy go as v squared?

Where does the squared come from?

The answer is kinematics

but to see that directly let's look at the quantities as functions of time.

Suppose that we start with an object at rest.

We supply a constant force for a certain time t.

Next we apply the force for twice as long, constant acceleration

for twice as long, so it ends up with twice the velocity.

But how far does it go?

For constant acceleration from rest the distance traveled goes as t squared.

So for a time twice as long we apply the force over four times the distance.

It takes four times as much work to double the speed.

Here's an important example with application to road safety.

I apply constant braking force to stop my bike.

180 degrees between braking force and displacement of the bike,

work done by the brakes is negative,

so kinetic energy decreases to zero.

If I am traveling twice as fast I have four times as much kinetic energy

so the same braking force needs four times the distance to stop me.

Please remember that on the road.

Remember what we've just done.

In one dimension, work equals total force times the distance.

But for constant acceleration, our kinematics equation three

said, "2a times distance traveled equals v squared minus v0 squared."

Substituting that gives work equals change in a half mV squared.

So we define half mv squared to be the kinetic energy and now we can say,

"Work done by all forces acting on the object equals increase in its kinetic energy.".

What we've proved is a theorem.

It's just a mathematical manipulation.

It's not a new law.

We defined work.

We then substituted from Newton's second law, F equals ma.

We used an equation from kinematics.

Finally, we defined kinetic energy.

The only physical law we used was F total equals ma.

So the Work-Energy Theorem is logically equivalent to Newton's second law.

If one is true, so is the other.

Falsify one, and you falsify both.

Nevertheless, the idea of energy is a powerful concept, pun intended, and it gives

us a very different way of looking at the world.

It's often helpful in doing problems and that must be a segue into a quiz.