This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 6.5: IMPLUSE FUNCTIONS KIAM HEONG KWA For a given tvalue, 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...
View Full
Document
 Winter '11
 Kwa
 Differential Equations, Equations

Click to edit the document details