limO (Llv/ Llt ) , or L\< �0' (12-9) OgmPh A 0' Substituting Eq.
12-17b. z The acceleration of the particle is obtained by taking the first time derivative of Eq. 12-11 (or the second time derivative of Eq. 12-10). We have Accele ration. ( 12-13) x /----- y Acceleration (c) where ax = Vx = x ay = Vy = Y az = Vz = Z (12-14) Here ax , ay , az represent, respectively, the first time derivatives of Vx = vAt ) , Vy = vy( t ) , Vz = vz(t), or the second time derivatives of the functions x = x(t), y = y(t), z = z(t). The acceleration has a magnitude a = Va� + a� + a� and a direction specified by the unit vector Da = ala .