Two particles having position vectors $\vec{r_{1}} = (3 \hat{i} - 5 \hat{j}) m$ and $\vec{r_{2}} = (- 5 \hat{i} - 3 \hat{j}) m$ are moving with velocities $\vec{V_{1}} = (4 \hat{i} - 4 \hat{j}) {ms}^{- 1}$ and $\vec{V_{2}} = (a \hat{i} - 3 \hat{j}) {ms}^{- 1} .$ If they collide after 2 seconds, the value of ‘a’

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answer is D.

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Detailed Solution

Condition for collision is
$t = \frac{|\bar{r_{2}} - \bar{r_{1}}|}{|\bar{v_{2}} - \bar{v_{1}}|} \Rightarrow 2 = \frac{|- 8 \hat{i} + 2 \hat{j}|}{|(a - 4) \hat{i} + \hat{j}|} \Rightarrow 2 = \frac{\sqrt{{(8)}^{2} + {(2)}^{2}}}{\sqrt{{(a - 4)}^{2} + {(1)}^{2}}} \Rightarrow 4 [{(a - 4)}^{2} + 1] = 64 + 4 \Rightarrow 4 [a^{2} + 16 - 8 a + 1] = 68 \Rightarrow a^{2} + 17 - 8 a = 17 \Rightarrow a^{2} = 8 a \Rightarrow a = 8$

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