Answer to Question #329030 in Calculus for asif

Question

The acceleration of a particle at any time is given by a = 12e

3t

i − 8sin2tj +

4tk. If the velocity is zero at t = 0, find velocity.

Answer 1

The acceleration of a particle at any time is given by

$\bold{a}=12e^{3t}\bold{i}-8\sin2t\bold{j} +4t\bold{k}$ .

Velocity:

$\bold v=\int \bold a dt=\int (12e^{3t}\bold{i}-8\sin2t\bold{j} +4t\bold{k})dt=\\ \frac{12}{3}e^{3t}\bold{i}+\frac 82\cos2t\bold{j} +\frac 42 t^2\bold{k}+C_i\bold{i}+C_j\bold{j} +C_k\bold{k}=\\ 4e^{3t}\bold{i}+4\cos2t\bold{j} +2 t^2\bold{k}+C_i\bold{i}+C_j\bold{j} +C_k\bold{k}$,

where $C_i,C_j,C_k$ are const.

$\bold v(0)=4\bold{i}+4\bold{j} +0\bold{k}+C_i\bold{i}+C_j\bold{j} +C_k\bold{k}=\bold 0$

$C_i=-4$ ; $C_j=-4$ ; $C_k=0$ .

Answer:

$\bold v=4(e^{3t}-1)\bold{i}+4(\cos2t-1)\bold{j} +2 t^2\bold{k}$