# Derivation Of Formulas For Energy

I had a tough time deriving formulas for mechanical energy when given in school as assignment or as a question in exams. I don’t want any of you reading this to experience the same feeling. So, I would like to show you how we derive the conventional formulas for potential and kinetic energies to be later used in problems.

## Potential Energy

We know that Work is equal to the product of the Force applied and the Distance travelled. So,

W = F x s

Also, Force is equal to the product of Mass of the body and the Acceleration. So,

W = ( m x a ) x s

Now, as the energy is being calculated on Earth, we have to substitute Acceleration with g ( acceleration due to gravity ) and the Distance with h ( height ). So,

W = ( m x g  ) x h

Therefore, we have derived our formula i.e.

W = m g h or EK = m g h

## Kinetic Energy

We know that Work is equal to the product of the Force applied and the Distance travelled. So,

W = F x s

Also, Force is equal to the product of Mass of the body and the Acceleration. So,

W = ( m x a ) x s

Now, the Distance can be substituted with ( v2 – u2 / 2 a ) from the third equation of motion ( v2 – u2 = 2 a s )

W = ( m x a ) ( v2 – u2 / 2 a )

W =  m ( v2 – u2 / 2 )

W = ½ m ( v2 – u2 )

Usually, the bodies start from rest i.e. u = 0. So,

W = ½ m v2

Therefore, we have derived our formula i.e.

W = ½ m v2 or EP = ½ m v2

I hope that I have been able to explain the steps to be followed while deriving the equations. 🙂