83.3 The acceleration in rectilinear motion 1.Average acceleration Define:ave== ave rave ∠t 2. Instantaneous acceleration if v;=v,(ti,v=v,(t+4t) Av=v-v,=lv(t+45)-v(x)i 83.3 The acceleration in rectilinear motion then as lim 4v= lim /(t+4t)-v2()i_dv() 4→0sAt→+0s dt or(= dv(t) dv(t): dr(t) d'x(t dt The instantaneous acceleration of a particle is the time rate of change of the velocity vector or the first derivative of the instantaneous velocity vector with respect to time; or the second derivative of the instantaneous position vector with respect to time5 §3.3 The acceleration in rectilinear motion Define: t v v t v a f i ∆ ∆ ∆ r r r r − ave = = 1. Average acceleration i t v t v t a a i x f x i x ˆ ( ) ( ) ˆ ave ave ∆ − = = r 2. Instantaneous acceleration if v v v v t t v x i v v t i v v t t i f i x x i x f x ˆ [ ( ) ( )] ˆ , ( ) ˆ ( ) = − = + − = = + ∆ ∆ ∆ r r r r r then t v t t v t t v t i t v a x x t t d d ( ) ˆ [ ( ) ( )] lim lim 0s 0s r r r = + − = = → → ∆ ∆ ∆ ∆ ∆ ∆ or i t x t t r t i t v t t v t a x ˆ d d ( ) d d ( ) d d ( ) d d ( ) 2 2 2 2 = = = = r r r r The instantaneous acceleration of a particle is the time rate of change of the velocity vector or the first derivative of the instantaneous velocity vector with respect to time;or the second derivative of the instantaneous position vector with respect to time. §3.3 The acceleration in rectilinear motion