derive the formula of kinetic energy and potential energy

Formula for kinetic energy:

Let a body of mass m moving with a velocity u.

After a time t due to a force F which produces acceleration a in the body and it changes the velocity of the body to v.

Thus, we have;

The initial velocity of the body = u

Final velocity of the body = v

The force applied on the body = F

Acceleration in the body = a

Let the displacement in the body is s, then,

The work done on the body = Force applied on the body $\times$ displacement in the body

W = F$\times$ s ........(1)

Using the second law of motion;

F= ma........(2)

And according to the third equation of motion;

$v^2=u^2+2as$

Thus,

$s=\frac{v^2-u^2}{2a}$ .........(3)

From equations (1), (2) and (3), we get;

$W=ma\times \frac{v^2-u^2}{2a}$

$W=\frac{1}{2}m(v^2-u^2)$

As the body is ibn motion; the work done on the body = Kinetic energy of the body

Thus,

$K=\frac{1}{2}m(v^2-u^2)$

For initial velocity(u) = 0

We get;

$K=\frac{1}{2}m{v}^2$

This is the formula for the kinetic energy of a body.

The formula for Potential energy:

The energy possessed by a body due to the position of the body is called the potential energy of the body.

Let a body of mass m is brought to a height of h,

Then work done

W= $weightofthebody\times height$

W=mgh

Since at height the body is kept therefore the work done to place the body at height h is the potential energy of the body.

Thus,

P=mgh

This is the formula for the potential energy of a body