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Three ants are sitting at the vertices of an equilateral triangle of side a. At t=0, ant 1 starts approaching ant 2 with a speed v, ant 2 starts approaching ant 3 with a speed v and ant 3 starts approaching ant 1 with a speed v. If a=1m, v=1m/s, match the following.
List-I List-II
P) The three ants meet at the centroid of the triangle at time (in s) (1) $\frac{\sqrt{3}}{2}$
Q) The angular velocity (in rad/s) of ant 2 with respect to ant 1 at t = 0 is (2) $\frac{25 \sqrt{3}}{256}$
R) The angular acceleration (in $r a d / s^{2}$ ) of ant 2 with respect to ant 1 at t = 0 is (3) $\frac{2}{3}$
S) The area of the equilateral triangle (in $m^{2}$ ) formed by the three ants at time t = a/4v will be (4) $\frac{3 \sqrt{3}}{4}$

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$P \to 4; Q \to 3; R \to 1; S \to 2$

$P \to 3; Q \to 1; R \to 1; S \to 4$

$P \to 1; Q \to 4; R \to 2; S \to 3$

$P \to 3; Q \to 1; R \to 4; S \to 2$

answer is B.

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Detailed Solution

$(P) t = \frac{3 a}{2 V} = \frac{2}{3} s (Q) ω_{0} = \frac{{(V_{⊥})}_{r e l}}{x} = \frac{\frac{\sqrt{3} V}{2}}{a} = \frac{\sqrt{3}}{2} (R) ω_{t} = \frac{{(V_{⊥})}_{r e l}}{x} = \frac{\sqrt{3} V}{2 x} α = \frac{d ω}{d t} = \frac{\sqrt{3} V}{2} (- \frac{1}{x^{2}}) (\frac{d x}{d t}) = \frac{\sqrt{3} V}{2 x^{2}} . (\frac{3 V}{2}) α_{0} = \frac{3 \sqrt{3}}{4}$

$(S) x = a - \frac{3 V}{2} t = a - \frac{3 V}{2} (\frac{a}{4 V}) = \frac{5 a}{8} A = \frac{1}{2} (b) (h) = \frac{1}{2} (x) (\frac{\sqrt{3} x}{2}) = \frac{\sqrt{3}}{4} {[\frac{5 a}{8}]}^{2} = \frac{25 \sqrt{3}}{256}$

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