Let's look at an example of an object that's falling.
So suppose, once again, I have an apple, as we've been using, at a particular height, h.
It has no kinetic energy. It's not moving.
And then I let go and it falls, now it picks up some kinetic energy.
Then it has a kinetic energy that it did not have before.
So we've somehow gone from a potential energy to a kinetic energy
as we saw when we were discussing energy and conservation of energy.
This is done by work.
So work is also able to be thought of as a way to change types of energy
or to increase or change your kinetic energy.
And this brings us to the work energy theorem,
which is basically saying that the energy,
the kinetic energy that you gain or the change in kinetic energy you have
in a problem like this one where the apple fell and changed its kinetic energy
is equal to the work done on the apple by the total sum of forces acting on the object.
And it is important that the change in kinetic energy
is equal to the sum of all the forces acting on your object.
So, if you have many forces in a problem all acting together,
you need to find the work done by each force
to find the total work done and that will be the change in the kinetic energy of your object.
So, let's make this more concrete and look at a particular example.
So we have a force of gravity pulling our object downwards.
It starts with zero velocity and then gains a downwards velocity at the end of its path.
From conservation of energy, we know that the kinetic energy
that the object has at the bottom of the path
will be equal to whatever the potential energy it had at the top of the path
because the total energy is conserved.
And as we saw at the top of the path, it's total energy is only potential energy
and at the bottom of the path because it's now zero,
it's only got kinetic energy and no potential energy.
So from conservation of energy,
the kinetic energy that our object has at the bottom of the path
will be equal to whatever potential energy it had at the top of the path
which in this case is m times g times h.
So the change in kinetic energy of our object
is what we're looking for in this green box here.
It's just whatever the final energy is minus whatever the initial energy is,
which in this case is mgh, the final energy minus zero, the initial energy.
And so somehow our object gained mgh of kinetic energy.
And by the way, we're not saying now that we have a new expression for kinetic energy
that is mgh instead of 1/2mv squared.
We found that the kinetic energy was equal in magnitude
to the quantity mgh using conservation of energy.
And so if this is the change in the kinetic energy of our object,
we can also find this change in kinetic energy using work instead.
So, we have a force being applied to this object and that force is downwards.
The distance that the object is moving is also downwards,
so we can use our work equation of force times distance
times the cosine of the angle between these two.
This angle is zero in this case and the reason for that is the force is acting directly downwards,
and the motion of the object is also directly downwards,
and so the angle between these 2 is zero.
They're pointing in exactly the same direction.
And so if the angle is zero, the cosine of zero is 1.
And so if we plug in this 1 to our work equation,
we have that the work is equal to the force, mg times the distance or the displacement,
which is h, times the angle between them which is the cosine of the angle between them, which is simply 1.
So, we get the full work equals mgh out of our work expression.
And so we can see as we hoped that from the green box,
we have that the change in kinetic energy of the object was mgh using conservation of energy.
We also saw from our work energy theorem that the work done on our object was mgh,
and that is, its change in kinetic energy and so this works.
We can flip it around and say what happens instead if the force is fighting against the direction of motion,
if it's trying to get the object to stop moving.
In this case, we could say maybe I throw the object upwards
and if the object is moving upwards while gravity is pulling downwards,
we have a different situation. So let's again first analyze this from the point of view
of conservation of energy and see what the change in kinetic energy is.
We know that initially the kinetic energy is 1/2mv squared so it's a non-zero kinetic energy initially,
but then finally, the object has a potential energy only, it has no kinetic energy.
From conservation of energy, we know that these are equivalent,
that the energy at the bottom of the path, the kinetic energy,
is equal to the energy it has at the top of the path, the potential energy, mgh,
and so we know that the change in potential energy, the change in energy rather
is equal to the difference between these two.
So, the change in kinetic energy is equal to zero minus mgh, zero is the final kinetic energy
at the top of the path, mgh we know from conservation of energy
is equal to in magnitude the amount of kinetic energy it originally had,
and so we know this object lost in magnitude mgh worth of kinetic energy.
Looking at it one more time in the perspective of work,
we can look at F times d times cosine of theta.
In this case we have the opposite situation
in which the object is moving upwards but the force of gravity is downwards
and so the angle between these two is 180 degrees
and so we take the cosine of 180 degrees which is negative 1 instead of positive 1.
So, you can see that the idea is the work is going to be a negative quantity.
We plug in our negative 1 and this is telling us that the force is fighting against your object,
it's not wanting it to go up in the air.
So your object is moving upwards while a force is pulling it downwards
and that's going to change the kinetic energy to be more negative or less kinetic energy.
Putting this all together, we now have that the work is equal to minus force,
mg times distance, h and again we get the exact same result
as we would have expected from our work kinetic energy theorem.
From conservation of energy, we have minus mgh and from work, we also have minus mgh.