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\newcommand\ds{\displaystyle}
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QUESTION

The velocity of a particle $P$ at time $t$ is given by $(\sin
t)\textbf{i}+t\textbf{j}$. Find the position of $P$ at time $t$
given that it is at the origin at time $t=0$.

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ANSWER

Velocity$\ds=\frac{d\mathbf{r}}{dt}=\sin t\mathbf{i}+t\mathbf{j}$
therefore\\ $\ds\textbf{r}=-\cos
t\mathbf{i}+\frac{t^2}{2}\mathbf{j}+\textbf{c}$\\ At $\ds t=0$,\
$\textbf{r}=\textbf{0}=-\mathbf{i}+\textbf{c}$ therefore
$\textbf{c}=\mathbf{i}$\\ Hence $\ds\textbf{r}=(1-\cos
t)\mathbf{i}+\frac{t^2}{2}\mathbf{j}$.




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