At time t second, a particle of mass 3kg has position vector $\overrightarrow{r}$ metre, where $\overrightarrow{r}=3\text{t}\hat{i}-4\cos \text{t}\hat{j}$ . The impulse of the force during the time interval $0\le \text{t}\le \frac{\pi }{2}$ is :            

$\frac{\text{dr}}{\text{dt}}=\frac{\text{3dt}}{\text{dt}}\hat{i}-4\frac{\text{d}}{\text{dt}}\cos \text{t}\hat{j}$

$\text{v}=3\hat{i}-4\left(-\sin \text{t}\right)\hat{j}$

$\text{v}=3\hat{i}+4\sin \text{t}\hat{j}$

$\text{v}\hat{i}\text{at}\text{t}=0$

$v\hat{i}=3\hat{i}$

$Pi=m\overline{v}i$

$=3\left(3\hat{i}\right)=9\hat{j}$

$\text{v}\hat{j}=3\hat{i}+4\sin \frac{\pi }{2}\hat{j}$

$v\hat{j}=3\hat{i}+4\hat{j}$

$P\hat{j}=3\left(3\hat{i}+4\hat{j}\right)$

$=9\hat{i}+12\hat{j}$

$\text{Impulse}=9\hat{i}+12\hat{j}-9\hat{i}$

$=12\hat{j}$