VC2.11

Given $a=\frac{dv}{dt}=12\cos 2t i-8\sin 2t j+16t k\,$ where a and v are acceleration and velocity vectors o fparticle at any time t.

Integrating (1), $v=6\sin 2t i+4\cos 2t j+8t^2 k+c\,$ [Equation 2]

But v=0 at t=0.

Therefore from (2) $c=-4j\,$ .

Hence by (2),the required velocity is given by $v=\frac{dr}{dt}=6\sin 2t i+4(\cos 2t-1)j+8t^2 k\,$ [Equation 3]

Integrating (3), $r=-3\cos 2t i+(2\sin 2t-4t)j+\frac{8}{3}t^3 k+d\,$ [Equation 4]

But r=0 at t=0. So from (4), $0=-3i+d,d=3i\,$

Therefore,by equation (4), the required displacement is given by $r=3(1-\cos 2t)i+2(\sin 2t-2t)j+\frac{8}{3}t^3 k\,$