CHAPTER 3 DIFFERENTIATION VECTORS OF d A 02 dt  12 12   Differentiating the second time ; w r t t d A 2   2 2t 2 sec2 tan iˆ t t k ˆ 2 t0 2 2 2 0kˆ    iˆ ˆj and d A 2 2 12 22 02 5     Example 3.3 A particle moves along a curve whose parametric equations are t , where is the time.  ,  2 cos 3t, ,  2 sin 3t x y z a) Determine its velocity and acceleration at any time. The position vector of the particle is r 2 cos 3tˆj 2 sin 3tkˆ r  xiˆ  yˆj  zkˆ  iˆ    6 sin 3tˆj  6 cos 3tkˆ iˆ d r  r dt and  18 cos 3tˆj 18 sin 3tkˆ iˆ d r 2 a  dt 2 t=0. b) Find the magnitudes of the velocity and acceleration at d r d r 2 t=0, At 6kˆ 18    and iˆ ˆj. dt dt 2 dt Then  t0 12 18 325   d r 2 dt 2 and 1 6   37 t0  d r
18 CHAPTER 3 DIFFERENTIATION OF VECTORS d A dt  12  02  12 Differentiating the second time w r t t ; d 2 A 2  iˆ  2 2t 2 ˆ 2 sec2 t tan t k 2 t0  iˆ  2 ˆj  0kˆ and d 2 A 2  12  22  02  5 Example 3.3 A particle moves along a curve whose parametric equations are x  , y  2 cos 3t, , z  2 sin 3t , where t is the time. a) Determine its velocity and acceleration at any time. The position vector r of the particle is r  xiˆ  yˆj  zkˆ  iˆ  2 cos 3tˆj  2 sin 3tkˆ r  d r dt  iˆ 6 sin 3tˆj  6 cos 3tkˆ and a  d 2 r dt 2  iˆ 18 cos 3tˆj 18 sin 3tkˆ b) Find the magnitudes of the velocity and acceleration at t=0. At t=0, d r dt   6kˆ and d 2 r dt 2  iˆ 18 ˆj. Then and d r dt d 2 r dt 2 t0  t0   12   37  325 1 6