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Three particles P,Q and R initially located at vertices of an equilateral triangle of side $l$ start moving with constant speed v each such that P always chases(moves towards) Q,Q always chases R and R always chases P. Trajectory followed by P is shown in the figure. Choose the correct alternative(s).

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Average velocity of P for its motion till the instant it catches Q is $(\frac{\sqrt{3}}{2}) v$

Average speed of P for its motion till it completes one revolution around centroid O is v.

Distance covered by P when it completes one revolution around centroid O of the triangle $\frac{2 l}{3} (1 - e^{- 2 \sqrt{3} π})$ .

Separation between P and Q as P completes one revolution around O is $l$ $e^{- (2 π) \sqrt{3}}$

answer is A, B, C, D.

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Detailed Solution

$- \frac{d r}{d t} = V \cos 30^{0} + \frac{r d θ}{d t} = V \sin 30^{0}$

$\Rightarrow - \frac{d r}{r d θ} = \sqrt{3}$

$\int_{l / \sqrt{3}}^{r} \frac{d r}{r} = - \sqrt{3} \int_{0}^{θ} d θ \Rightarrow \frac{r}{l / \sqrt{3}} = e^{- \sqrt{3} θ}$

$\Rightarrow r = \frac{l}{\sqrt{3}} e^{- \sqrt{3} θ}; d r = \frac{l}{\sqrt{3}} e^{- \sqrt{3} θ} (- \sqrt{3} d θ)$

$d s = \sqrt{d r^{2} + {(\frac{d r}{\sqrt{3}})}^{2}} = \frac{2}{\sqrt{3}} | d r | = \frac{2}{\sqrt{3}} (- l) e^{- \sqrt{3} θ} d θ$

$s = + \frac{2 l}{\sqrt{3}} \int_{0}^{2 π} e^{- \sqrt{3} θ d θ} = + \frac{2 l}{\sqrt{3}} {(\frac{e^{- \sqrt{3} θ}}{- \sqrt{3}})}_{0}^{2 π}$

$= \frac{2 l}{3} (- e^{- \sqrt{3} (2 π)} + 1) = \frac{2 l}{3} (1 - e^{- \sqrt{3} (2 π)})$

Avg. Velocity of P = $= \frac{l / \sqrt{3}}{t}, t = \frac{l}{\frac{3 v}{2}} = \sqrt{3} v / 2$ .

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