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Derive kinetic energy by calculus method\xa0 |
| Answer» Ans.\xa0When 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\xa0F\xa0=\xa0ma\xa0The work done by the force will become converted into increased kinetic energy in the body.\xa0Derivation Using Calculus\xa0:Begin with the Work-Energy Theorem :\xa0The work that is done on an object is related to the change in its kinetic energy\xa0∆K\xa0=\xa0WRewrite work as an integral:\xa0we can represent the work done\xa0in terms of a velocity differential.∆K\xa0=\xa0∫F.\xa0drRewrite force in terms of velocity:\xa0mass is a scalar and can therefore be factored out.\xa0∆K\xa0=\xa0∫ma.dr=>\xa0∆K\xa0=\xa0m∫a.dr=>\xa0∆K\xa0=\xa0m∫dvdt.dr=>\xa0∆K\xa0=\xa0m∫drdt.dv=>\xa0∆K\xa0=\xa0m∫v.dv=>\xa0∆K\xa0=\xa012mv2\xa0\xa0 | |