Question

derive the formula for the kinetic energy of an object of mass m moving with velocity v

Answer 1

$\frac{1}{2} mv ^{2}$

Answer 2

We know that Kinetic energy is the work done by an object in motion.

=> Kinetic Energy = Work Done = Force * Displacement

We know that Displacement ( s ) can be taken from the third law of equation of motion.

Let the initial object be ' u ', final velocity be ' v ' and acceleration of that object be ' a '. The distance covered by object is ' s '

=> Third law of motion = v² - u² = 2as

=> s = v² - u² / 2a ( Transposing 2a to the RHS ) => Equation 1

Force = Mass * Acceleration

=> Force ( F ) = ma => Equation 2

Work Done = Force * Displacement

=> W = F * s

Substituting the formula from Equation 1 and 2, we get,

W = ma * ( v² - u² / 2a )

' a ' gets cancelled. We get,

W = m * ( v² - u² / 2 )

u = 0 . Since object starts from rest. Hence u² = 0² = 0.

=> W = m * v² / 2

=> W = 1 / 2 * m * v²

=> W = 1 / 2 mv²

Work Done is Kinetic energy.

=> Kinetic energy = 1 / 2 mv²

Hence derived !