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Unformatted text preview: 6.5: IMPLUSE FUNCTIONS KIAM HEONG KWA For a given t-value, say t , consider the family of functions (1) ( t- t ) = ( 1 2 if t- < t < t + , if t t- or t t + , that are indexed by > 0. These functions are known as impulse functions centered at t . Physically, each of the impulse functions can be thought of as a force that acts over the time interval ( t- ,t + ). As a consequence of Newtons second law, the integral (2) Z - ( t- t ) dt = 1 represents the net change in momentum of an object to which the force is applied. It is usually known as the impulse of the force over the interval ( t- ,t + ). In fact, the identity (2) is a special case of (3) lim Z - f ( t ) ( t- t ) dt = f ( t ) , where f is a continuous function on R . This is so since for each > 0, there is a constant [- , ] such that (4) Z - f ( t ) ( t- t ) dt = 1 2 Z t + t- f ( t ) dt = f ( t + ) by the mean value theorem. Now if we let 0, then 0, from which (3) follows in view of the continuity of...
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