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  4. A particle moves so that its position vector varies with time as vecr=A cos ω t hati+A sin ω t hatj. The initial velocity of the particle is

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Q. A particle moves so that its position vector varies with time as $\vec{r}=A \cos \omega t \hat{i}+A \sin \omega t \hat{j}$. The initial velocity of the particle is

Motion in a Plane

Solution:

Substituting $\vec{r}=(A \cos \omega t \hat{i}+A \sin \omega t \hat{j})$ in $\vec{v}=\frac{d \vec{r}}{d t}$ we have
$\vec{v} =A \frac{d}{d t}(\cos \omega t) \hat{i}+A \frac{d}{d t}(\sin \omega t) \hat{j}$
$=-A \omega \sin \omega t \hat{i}+A \omega \cos \omega t \hat{j}$
At $t=0, v=-A \omega \sin 0 \hat{i}+A \omega \cos 0 \hat{j}=A \omega \hat{j}$