Derive an expression for k.e of a body of mass m moving with a velocity v by calculus method.

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Derive kinetic energy by calculus method

Posted by Manvendra Kanojiya 5 years, 8 months ago

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 = ∫F. dr

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

∆K = ∫ma.dr

=> ∆K = m∫a.dr

=> ∆K = m∫dvdt.dr

=> ∆K = m∫drdt.dv

=> ∆K = m∫v.dv

=> ∆K = 12mv2