VC2.11

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

Integrating (1), v=6\sin 2ti+4\cos 2tj+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 2ti+4(\cos 2t-1)j+8t^{2}k\, [Equation 3]

Integrating (3),r=-3\cos 2ti+(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\,

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