# Types of Energy

- Derivation of kinetic energy
- Derivation of potential energy

## Kinetic Energy

Kinetic energy is the energy possessed by an object because of its motion. For example, a fast moving pebble can injure a person or break glass pane of window. Energy of moving vehicle, a fast moving wind can damage many house, or wind can move blades of wind mill, etc. All of this is possible because of kinetic energy.

### Drivation of Kinetic Energy

Suppose, the mass of a moving object = m

The initial velocity of a moving object = u

The acceleration of the object = a

The final velocity of the object = v

Displacement of object to achieve the final velocity = s.

We know from the equation of motion that,

*v ^{2} = u^{2} + 2as*

Or, *2as = v ^{2} - u^{2}*

Or, `s=(v^2-u^2)/(2a)` -----(i)

Now, we know that, Work done = *W = F × s*

By substituting the value of *s* from equation (i) in the expression *W = F × s* we get

`W=Fxx(v^2-u^2)/(2a)`

Now, according to Newtonâ€™s Second Law of motion, *Force = mass × acceleration*

Or, *F = m × a*

Therefore, by substituting the value of F in equation (ii) we get,

`W=m×a×(v^2-u^2)/(2a)`

`=>W=1/2 m(v^2-u^2)` ---(iii)

If the object starts moving from the state of rest, then initial velocity (u) will be equal to zero.

Therefore, equation (iii) can be written as

`=>W=1/2m(v^2-0^2)`

`=>W= 1/2 mv^2` ------(iv)

Equation (iv) shows that work done is equal to the change in kinetic energy of an object.

Therefore, if an object of mass *m* is moving with a constant velocity,

The Kinetic Energy `(E_k) = 1/2 mv^2` ----(v)

From the above equation it is clear that kinetic energy of a moving object increases with increase of mass and velocity of the object.

## Potential Energy

Energy possessed by an object because of its position is called potential energy. For example, when a stone is kept at a height, it possesses some energy because of its height. Because of this potential energy, object kept at a height falls over the ground.

A stretched rubber band possesses some energy because of its position. Because of that energy, when the stretched rubber band is released it acquires its original position by movement. A stretched catapulted possesses potential energy because of its stretched string and is able to do some work.

A stretched bow possesses energy because of its position of stretched string.

### Expression for Potential Energy:

Potential energy possessed by an object due to its height

Let and object of mass *m* is placed over a height, h against gravity.

Therefore, the minimum force required to work done, *F = mg*

Where, *F* is force, *m* is mass and *g* is the acceleration due to gravity.

We know that, work done = *Force × displacement*

Therefore, Work done, *W = F × h*

Where, *h* is the displacement of the object. Since, the object is displaced at a height, therefore, *h* is taken at the place of *s*.

Or, *W = mg h* (since, *F = mg*)

The potential energy (E_{p}) is equal to the work done over the object

Therefore, *E _{p} = mgh*

The potential energy of an object depends upon the mass and height (position) of the object and not upon the path.