Derive kinetic energy by calculus method …

Ans. When there are no opposing forces, a moving body tends to keep moving with a steady velocity as we know from Newton's first law of motion. If, however, a resultant force does act on a moving body in the direction of its motion, then it will accelerate per Newton's second law $F = ma$ The work done by the force will become converted into increased kinetic energy in the body. 

Derivation Using Calculus :

Begin with the Work-Energy Theorem : The work that is done on an object is related to the change in its kinetic energy<meta aria-hidden="true" />

$∆K = W$

Rewrite work as an integral: we can represent the work done in terms of a velocity differential.

$∆K = \int F. dr$

Rewrite force in terms of velocity: mass is a scalar and can therefore be factored out.

$∆K = \int ma.dr$

=> $∆K = m\int a.dr$

=> $∆K = m\int \frac{dv}{dt}.dr$

=> $∆K = m\int \frac{dr}{dt}.dv$

=> $∆K = m\int v.dv$

=> $∆K = \frac{1}{2}m{v}^2$

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