Work done and Energy Transfers

Derivation of the equations for gravitational Potential energy and also Kinetic energy:

Gravitational Potential Energy:

We begin with the equation for work done;

Work\ Done = Fd

Since gravitational potential energy only changes if the height of the object is altered, the distance, d, must be solely related to the height, h. So

Work\ Done = Fh\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (1)

Now, any object that has a mass will have a weight, which is a force and acts downwards towards Earth. This can be written in the equation, F = mg

So, substituting this into equation (1) gives us:

Work\ Done = mgh

Since work done is just energy, we can say that this type of energy is the gravitational potential energy:

G.P.E = mgh


Kinetic Energy

We begin with the equation for work done;

Work\ Done = Fd

In this case we can substitute in Newton’s 2nd law of motion; F = ma

Work\ Done = mas\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (1)

We then use SUVAT and appropriate values for each quantity;
s = s
u = 0
v = v
a = a (we assume negligible air resistance)
t = X

Therefore we use the equation v^2 = u^2 + 2as
Subbing the value from above in gives:
v^2 = 2as
\frac{1}{2}v^2 = as

Subbing this into equation (1) gives:

Work\ Done = mas = m\frac{1}{2}v^2

Since the object has a velocity, we can say that the work done the transfer into kinetic energy:

Kinetic\ Energy = \frac{1}{2}mv^2

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