Q. 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 $vec{r_1}$ respectively. The condition for particles A and B for their collision is

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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 $vec{r_1}$ respectively.
The condition for particles A and B for their collision is

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 … (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 $frac{vec{r_1}-vec{r_2}}{|vec{r_1}-vec{r_2}|}=frac{(vec{v_2}-vec{v_1})}{|vec{v_2}-vec{v_1}|}$