Two particles having position vectors r→1=(3i^+5j^)m and r→2=(−5i^+3j^)m are moving with velocities V→1=(4i^−4j^)ms−1 and V→2=(ai^−3j^) ms−1. If they collide after 2 seconds, the value of 'a' is

Condition for collision ist=r¯1−r¯2v¯2−v¯1⇒2=|8i^+2i^||(a−4)i^+j^|⇒2=(8)2+(2)2(a−4)2+(1)2⇒4(a−4)2+1=64+4⇒4a2+16−8a+1=68;⇒a2+17−8a=17⇒a2=8a⇒a=8

Two particles having position vectors r→1=(3i^+5j^)m and r→2=(−5i^+3j^)m are moving with velocities V→1=(4i^−4j^)ms−1 and V→2=(ai^−3j^) ms−1. If they collide after 2 seconds, the value of 'a' is

Condition for collision ist=r¯1−r¯2v¯2−v¯1⇒2=|8i^+2i^||(a−4)i^+j^|⇒2=(8)2+(2)2(a−4)2+(1)2⇒4(a−4)2+1=64+4⇒4a2+16−8a+1=68;⇒a2+17−8a=17⇒a2=8a⇒a=8

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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' is

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answer is D.

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Detailed Solution

Condition for collision is
$\begin{array}{l} t = \frac{|{\bar{r}}_{1} - {\bar{r}}_{2}|}{|{\bar{v}}_{2} - {\bar{v}}_{1}|} \Rightarrow 2 = \frac{| 8 \hat{i} + 2 \hat{i} |}{| (a - 4) \hat{i} + \hat{j} |} \\ \Rightarrow 2 = \frac{\sqrt{(8)^{2} + (2)^{2}}}{\sqrt{(a - 4)^{2} + (1)^{2}}} \\ \Rightarrow 4 [(a - 4)^{2} + 1] = 64 + 4 \\ \Rightarrow 4 [a^{2} + 16 - 8 a + 1] = 68; \\ \Rightarrow a^{2} + 17 - 8 a = 17 \\ \Rightarrow a^{2} = 8 a \Rightarrow a = 8 \end{array}$