A particle is moving in a circular path of radius $1 m$ with a constant angular velocity of $π rad / s$ in anticlockwise direction. The circle is in $x y$ -plane with center at origin. At $t = 0$ ,the particle is at $(1, 0)$

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its velocity at $t = \frac{1}{3}$ sec is $\frac{π}{2} \hat{i} + \frac{\sqrt{3 π}}{2} \hat{j} m / s$

its acceleration at $t = \frac{1}{3}$ sec is $- π^{2} [\frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j}] m / s^{2}$

its velocity at $t = \frac{1}{3}$ sec is $\frac{- π \sqrt{3}}{2} \hat{i} + \frac{π}{2} \hat{j} m / s$

its acceleration at $t = \frac{1}{3}$ sec is $- π^{2} [\frac{1}{2} \hat{i} + \frac{\sqrt{3}}{2} \hat{j}] m / s^{2}$

answer is B, C.

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Detailed Solution

$t = \frac{1}{3}, θ = ω t = π \times \frac{1}{3} = \frac{π}{3}, \vec{v} = π \hat{k} \times [1 \cos \frac{π}{3} \hat{i} + 1 \sin \frac{π}{3} \hat{j}] = \frac{π \sqrt{3}}{2} (- \hat{i}) + \frac{π}{2} \hat{j} \vec{a} = - ω^{2} \vec{r} = - π^{2} [1 \cos \frac{π}{3} \hat{i} + 1 \sin \frac{π}{3} \hat{j}] = - π^{2} [\frac{\hat{i}}{2} + \frac{\sqrt{3}}{2} \hat{j}]$