Question

A particle moves such that its position vector $\vec{r}\left(t\right)=cos\omega\,t\,\hat{i} + sin\,\omega t\,\hat{j}$ where $\omega$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $\vec{v}\left(t\right)$ and acceleration $\vec{a}\left(t\right)$ of the particle :

• A. $\vec{v}$ and $\vec{a}$ both are parallel to $\vec{r}$
• B. $\vec{v}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed away from the origin
• C. $\vec{v}$ and $\vec{a}$ both are perpendicular to $\vec{r}$
• D. $\vec{v}$ is perpendicular to $\vec{r}$ and $\vec{a}$ is directed towards the origin

Answer 1

Answer: (D)

$\vec{r}\left(t\right)=cos\,\omega\hat{i}+sin\,\omega t\,\hat{j}$
On diff. we get
$\vec{v}=-\omega\,sin\,\omega t\,\hat{i}+\omega\,cos\,\omega t\,\hat{j}$
$\vec{a}=-\omega^{2}\vec{r}$
$\vec{v}. \vec{r}=0$