# 6.5we - 6.5 IMPLUSE FUNCTIONS KIAM HEONG KWA For a given...

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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) δ ( t- t ) = ( 1 2 if t- < t < t + , if t ≤ t- or t ≥ t + , that are indexed by > 0. These functions δ 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 Newton’s second law, the integral (2) Z ∞-∞ δ ( t- t ) dt = 1 represents the net change in momentum of an object to which the force δ 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 ) δ ( t- t ) dt = f ( t ) , where f is a continuous function on R . This is so since for each > 0, there is a constant σ ∈ [- , ] such that (4) Z ∞-∞ f ( t ) δ ( t- t ) dt = 1 2 Z t + t- f ( t ) dt = f ( t + σ ) by the mean value theorem. Now if we let → 0, then σ → 0, from which (3) follows in view of the continuity of...
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6.5we - 6.5 IMPLUSE FUNCTIONS KIAM HEONG KWA For a given...

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