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

Moderate

A

${\vec{r}}_{1} \times {\vec{v}}_{1} = {\vec{r}}_{2} \times {\vec{v}}_{2}$
B

${\vec{r}}_{1} - {\vec{r}}_{2} = {\vec{v}}_{1} - {\vec{v}}_{2}$
C

$\frac{{\vec{r}}_{1} - {\vec{r}}_{2}}{|{\vec{r}}_{1} - {\vec{r}}_{2}|} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{|{\vec{v}}_{2} - {\vec{v}}_{1}|}$
D

${\vec{r}}_{1} \cdot {\vec{v}}_{1} = {\vec{r}}_{2} \cdot {\vec{v}}_{2}$

Solution

Let the particles A and B collide at time t. For their collision, the position vectors of both particles should be same at time t, i.e.
${\vec{r}}_{1} + {\vec{v}}_{1} t = {\vec{r}}_{2} + {\vec{v}}_{2} t$
${\vec{r}}_{1} - {\vec{r}}_{2} = {\vec{v}}_{2} t - {\vec{v}}_{1} t$
$= ({\vec{v}}_{2} - {\vec{v}}_{1}) t \dots (i)$
$Also, |{\vec{r}}_{1} - {\vec{r}}_{2}| = |{\vec{v}}_{2} - {\vec{v}}_{1}| t$
$or t = \frac{|{\vec{r}}_{1} - {\vec{r}}_{2}|}{|{\vec{v}}_{2} - {\vec{v}}_{1}|}$
$Substituting this value of t in eqn. (i), we get$
${\vec{r}}_{1} - {\vec{r}}_{2} = ({\vec{v}}_{2} - {\vec{v}}_{1}) \frac{|{\vec{r}}_{1} - {\vec{r}}_{2}|}{|{\vec{v}}_{2} - {\vec{v}}_{1}|} or \frac{{\vec{r}}_{1} - {\vec{r}}_{2}}{|{\vec{r}}_{1} - {\vec{r}}_{2}|} = \frac{({\vec{v}}_{2} - {\vec{v}}_{1})}{|{\vec{v}}_{2} - {\vec{v}}_{1}|}$

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