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Unformatted text preview: Kinetic Energy, Work, and Power Shina Tan Kinetic energy of a single particle The -th particle in a system of n particles obeys Newtons Second Law m r ( t ) = F ( t ) , (1) where F ( t ) = F ( e ) ( t ) + f ( t ), F ( e ) ( t ) is the force exterted by objects external to the system, and f ( t ) is the force exterted by the -th particle in the system. Taking the scalor products of both sides of Eq. (1) with the instantaneous velocity v ( t ) = r ( t ) , and noting that d dt h 1 2 m v 2 ( t ) i = d dt h 1 2 m v ( t ) v ( t ) i = 1 2 m v ( t ) v ( t ) + 1 2 m v ( t ) v ( t ) = m v ( t ) v ( t ) = m r ( t ) v ( t ) , we get d dt h 1 2 m v 2 ( t ) i = F ( t ) d r ( t ) dt . The left hand side is the rate of change of the kinetic energy, and the right hand side F ( t ) v ( t ) is called the power exerted on the particle. Integrating this equation over time from t 1 to t 2 , we get 1 2 m v 2 ( t 2 )- 1 2 m v 2 ( t 1 ) = Z t 2 t 1 F ( t ) d r ( t ) . (2) The left hand side is the change of the kinetice energy 1 2 m v 2 , and the right hand side is the work done on the particle. 1 Kinetic energy of a system of particles Definition 1 The kinetic energy of a system of particles is defined as T ( t ) = X 1 2 m v 2 ( t ) . (3)...
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