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Relative Velocity in 2D

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Question

Two particles 'A' and 'B', move with constant velocities $_{1}$ and $_{2}$ respectively. At the initial moment, their position vectors are $_{1}$ and $_{2}$ respectively. The condition for particle 'A' and 'B' for their collision is

A

- \to r_{1} -_{2} =_{1} -_{2}

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B

\frac{- \to r_{1} -_{2}}{(- \to r_{1} -_{2} ∣ ∣} = \frac{_{1} -_{2}}{(_{1} -_{2} ∣ ∣}

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C

_{1} ._{1} =_{2} ._{2}

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D

_{1} \times_{1} =_{2} \times_{2}

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Solution

The correct option is B $\frac{- \to r_{1} -_{2}}{(- \to r_{1} -_{2} ∣ ∣} = \frac{_{1} -_{2}}{(_{1} -_{2} ∣ ∣}$
For two particles to collide, the direction of the relative velocity of one with respect to other should be directed towards the relative position of one particle wrt other.
i.e. $\frac{- \to r_{1} -_{2}}{(- \to r_{1} -_{2} ∣ ∣}$ $\to$ direction of relative position of 1 w.r.t. 2
& $\frac{_{1} -_{2}}{(_{1} -_{2} ∣ ∣}$ $\to$ direction of velocity of 2 w.r.t. 1

So for collision of 'A' & 'B',
$\frac{- \to r_{1} -_{2}}{(- \to r_{1} -_{2} ∣ ∣} = \frac{_{1} -_{2}}{(_{1} -_{2} ∣ ∣}$

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