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 their collision is
  • A
    ${\vec r_1} - {\vec r_2} = {\vec v_1} - {\vec v_2}$
  • $\frac{{{{\vec r}_1} - {{\vec r}_2}}}{{\left| {{{\vec r}_1} - {{\vec r}_2}} \right|}} = \frac{{{{\vec v}_2} - {{\vec v}_1}}}{{\left| {{{\vec v}_2} - {{\vec v}_1}} \right|}}$
  • C
    ${{\vec r}_1}.{{\vec v}_1} = {{\vec r}_2}.{{\vec v}_2}$
  • D
    ${{\vec r}_1} \times {{\vec v}_1} = {{\vec r}_2} \times {{\vec v}_2}$

Answer

Correct option: B.
$\frac{{{{\vec r}_1} - {{\vec r}_2}}}{{\left| {{{\vec r}_1} - {{\vec r}_2}} \right|}} = \frac{{{{\vec v}_2} - {{\vec v}_1}}}{{\left| {{{\vec v}_2} - {{\vec v}_1}} \right|}}$
b
For two particles to collide, the direction of the relative velocity of one with respect to other should be directed towands the relative position of the other particle

ie. $\frac{\vec{\mathrm{r}}_{1}-\vec{\mathrm{r}}_{2}}{\left|\vec{\mathrm{r}}_{1}-\vec{\mathrm{r}}_{2}\right|} \rightarrow$ clivection of relative position of $1 \mathrm{w.r.t}$ $2$

 $\frac{\vec{\mathrm{v}}_{2}-\vec{\mathrm{v}}_{1}}{\left|\vec{\mathrm{v}}_{2}-\vec{\mathrm{v}}_{1}\right|} \rightarrow$ direction of velocity of $2 \mathrm{w}$ $r.t. 1$

so for collision of $\mathrm{A} \& \mathrm{B}$

$\frac{\overrightarrow{\mathrm{r}}_{1}-\overrightarrow{\mathrm{r}}_{2}}{\left|\overrightarrow{\mathrm{r}}_{1}-\overrightarrow{\mathrm{r}}_{2}\right|}=\frac{\overrightarrow{\mathrm{v}}_{2}-\overrightarrow{\mathrm{v}}_{1}}{\left|\overrightarrow{\mathrm{v}}_{2}-\overrightarrow{\mathrm{v}}_{1}\right|}$

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