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Unformatted text preview: J. Peraire Dynamics 16.07 Fall 2004 Version 1.1 REVIEW - Rectilinear Motion We start by considering the simple motion of a particle along a straight line. The position of particle A at any instant can be specified by the coordinate s with origin at some fixed point O . The instantaneous velocity is ds v = dt = ˙ v . (1) We will be using the “dot” notation, to indicate time derivative, e.g. (˙) ≡ d/dt . Here, a positive v means that the particle is moving in the direction of increasing s , whereas a negative v , indicates that the particle is moving in the opposite direction. The acceleration is dv d 2 s ¨ a = = v ˙ = = s . (2) dt dt 2 The above expression allows us to calculate the speed and the acceleration if s and/or v are given as a function of t , i.e. s ( t ) and v ( t ). In most cases however, we will know the acceleration and then, the velocity and the position will have to be determined from the above expressions by integration. Determining the velocity from the acceleration From a ( t ) If the acceleration is given as a function of t , a ( t ), then the velocity can be determined by simple integration of equation (2), t v ( t ) = v 0 + a ( t ) dt . (3) t 0 Here, v 0 is the velocity at time t , which is determined by the initial conditions. From a ( v ) If the acceleration is given as a function of velocity a ( v ), then, we can still use equation (2), but in this case we will solve for the time as a function of velocity, v dv t ( v ) = t 0 + . (4) a ( v ) v 0 1 Once the relationship t ( v ) has been obtained, we can, in principle, solve for the velocity to obtain v ( t ). )....
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This note was uploaded on 01/12/2012 for the course ENG 102 taught by Professor Eke during the Winter '08 term at UC Davis.
- Winter '08