Two particles A and B, move with constant velocities v_1 and v_2. At the initial moment their position vectors are r_1 and r_2 respectively. The condition for particles A and B for their collision is

A. r_1 - r_2 | r_1 - r_2 | = ; v_2 - v_1 | v_2 - v_1 | ; ; ;

Two particles A and B, move with constant velocities v_1 and v_2.…

MCQ

Two particles $A$ and $B,$ move with constant velocities $\vec v_1$ and $\vec v_2$. 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

✓
$\frac{{\overrightarrow {{r_1}} - \overrightarrow {{r_2}} }}{{\left| {\overrightarrow {{r_1}} - \overrightarrow {{r_2}} } \right|}} = \;\frac{{\overrightarrow {{v_2}} - \overrightarrow {{v_1}} }}{{\left| {\overrightarrow {{v_2}} - \overrightarrow {{v_1}} } \right|}}\;\;\;$
B
$\overrightarrow {{r_1}} $ -$\overrightarrow {{r_2}} $ = $\overrightarrow {{v_1}} $ -$\overrightarrow {{v_2}} $
C
$\;\overrightarrow {{r_1}} $ .$\;\overrightarrow {{v_1}} $ =$\overrightarrow {{r_2}} $ .$\;\overrightarrow {{v_2}} $
D
$\;\overrightarrow {{r_1}} \times \overrightarrow {{v_1}}=\overrightarrow {{r_2}} \times \overrightarrow {{v_2}} $

✓

Answer

Correct option: A.

$\frac{{\overrightarrow {{r_1}} - \overrightarrow {{r_2}} }}{{\left| {\overrightarrow {{r_1}} - \overrightarrow {{r_2}} } \right|}} = \;\frac{{\overrightarrow {{v_2}} - \overrightarrow {{v_1}} }}{{\left| {\overrightarrow {{v_2}} - \overrightarrow {{v_1}} } \right|}}\;\;\;$

a
$\begin{array}{l}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Let\,the\,particles\,A\,and\,B\,collide\\
at\,time\,t.\,For\,their\,collision,\,the\,position\\
vectors\,of\,both\,particles\,sholud\,be\,\\
same\,at\,time\,t,\,i.e.,\\
{{\vec r}_1} + {{\vec v}_1}t = {{\vec r}_2} + {{\vec v}_2}t\,;\,{{\vec r}_1} - {{\vec r}_2} = {{\vec v}_2}t - {{\vec v}_1}t\\
= \left( {{{\vec v}_2} - {{\vec v}_1}} \right)t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)
\end{array}$

$\begin{array}{l}
Also,\,\left| {\,{{\vec r}_1} - {{\vec r}_2}} \right| = \left| {{{\vec v}_2} - {{\vec v}_1}} \right|t\,\,or\,t = \frac{{\left| {\,{{\vec r}_1} - \vec r} \right|}}{{\left| {{{\vec v}_2} - {{\vec v}_1}} \right|}}\\
Substituting\,this\,value\,of\,t\,in\,eqn.\,\left( i \right),\\
we\,get\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\vec r}_1} - {{\vec r}_2} = \left( {{{\vec v}_2} - {{\vec v}_1}} \right)\frac{{\left| {\,{{\vec r}_1} - \vec r} \right|}}{{\left| {{{\vec v}_2} - {{\vec v}_1}} \right|}}\\
or\,\frac{{\left| {\,{{\vec r}_1} - {{\vec r}_2}} \right|}}{{\left| {\,{{\vec r}_1} - {{\vec r}_2}} \right|}} = \frac{{\left( {{{\vec v}_2} - {{\vec v}_1}} \right)}}{{\left| {{{\vec v}_2} - {{\vec v}_1}} \right|}}
\end{array}$