Question

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

• A. $\vec{r_1} \times \vec{v_1}=\vec{r_2} \times \vec{v_2}$
• B. $\vec{r_1} -\vec{r_2}=\vec{v_1} \times \vec{v_2}$
• C. $\frac{\vec{r_1}-\vec{r_2}}{|\vec{r_1}-\vec{r_2}|}=\frac{\vec{v_1}-\vec{v_2}}{|\vec{v_1}-\vec{v_2}|}$
• D. $\vec{r_1} \cdot \vec{v_1}=\vec{r_2} \cdot \vec{v_2}$

Answer 1

Answer: (C)

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 ... (i) $
Also, $|\vec{r_1}-\vec{r_2}|=|\vec{v_1}-\vec{v_1}|t$
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 $\frac{\vec{r_1}-\vec{r_2}}{|\vec{r_1}-\vec{r_2}|}=\frac{(\vec{v_2}-\vec{v_1})}{|\vec{v_2}-\vec{v_1}|}$