**Derivation of Equation for Kinetic Energy**

**In case of translational motion: Amount of work done by a moving body before coming to rest is the measure of kinetic energy.**

Suppose, a body of mass ‘m’ is moving along AB with a velocity v. A constant force F is applied opposite to the motion along BA. Due to this uniform retardation will be produced. Let the uniform retardation = a and the body came to rest at point B after covering a distance s from point A. Here, final velocity. v = 0.

**Kinetic energy = work done before coming to rest**

= force x distance travelled before coming to rest = F x s

We know from Newton’s second law of motion;

Force = mass x acceleration or retardation

means, **F = ma**

According to description, 0 = v^{2} – 2as

or, 2as = v^{2}; or, s = v^{2}/2a

Putting the values of F and s in the above equation, we get,

Kinetic energy = ma x v^{2}/2a = ½ mv^{2}

or, **K. E. = ½ mv ^{2}**

That means, **kinetic energy (K.E.) = ½ mv ^{2} = ½ mass x (velocity)^{2}**

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