# Work and Energy

## Table of Contents

## Kinetic Energy

Kinetic energy is the energy possessed by an object because of 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.

### Kinetic Energy Possessed by a moving object

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`

`=>2as=v^2-u^2`

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

Now, we know that, Work done, `W = F xx s`

Thus, by substituting the value of ‘s’ from equation (i) in the expression W = F x s, we get

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

Now, according to Newton’s Second Law of motion, Force = mass x acceleration

Or, F = m x a

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

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

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

If the object starts moving from the state of rest, therefore, 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,

Thus, 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 x displacement

Therefore, Work done, W = F x 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

Where, ‘h’ is height, ‘m’ is mass and ‘g’ is acceleration due to gravitiy.

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