Question:
$\text{Two particles A and B, move with constant velocities } \vec{v_1} \text{ and } \vec{v_2}\text{. At the initial moment their position vector are } \vec{r_1} \text{ and } \vec{r_2} \text{ respectively. The conditions for particles A and B for their collision to happen will be:}$
Solution:
\text{Hint: For collision, their relative velocities should direct towards each other.}
\text{Step 1: Find the relative position of one particle w.r.t. the other particle.}
\text{For two particles A and B moving with constant velocities } v_1 \text{ and } v_2\text{,}
\text{such that two particles collide, the direction of the relative velocity of one}
\text{with respect to the other should be directed towards the relative position}
\text{of the other particle.}
\text{Direction of relative position of 1 w.r.t. 2 = } \frac{\vec{r_1} - \vec{r_2}}{|\vec{r_1} - \vec{r_2}|}
\text{Step 2: Find the direction of the velocity of one particle w.r.t. the other particle.}
\text{Similarly, direction of the velocity of 1 w.r.t. 2 = } \frac{\vec{v_1} - \vec{v_2}}{|\vec{v_1} - \vec{v_2}|}
\text{Step 3: Find the condition of collision.}
\text{So, for collision of A and B, we get:}
\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}|}
\text{Alternate Method:}
\text{Step 1: Find the displacement of each particle.}
\text{As the resultant displacement of the particles:}
R = \vec{r_1} + \vec{v_1}t
R' = \vec{r_2} + \vec{v_2}t
\text{Step 2: Find the condition of collision.}
\text{For collision, } R = R'
\text{Therefore: } \vec{r_1} + \vec{v_1}t = \vec{r_2} + \vec{v_2}t
\text{This gives us: } \vec{r_1} - \vec{r_2} = (\vec{v_2} - \vec{v_1})t \text{ ...(1)}
\text{Step 3: As the options don't have the time variable, we need to find}
\text{a clever way to get rid of t.}
\text{As equal vectors have equal magnitude, from (1), we can take}
\text{absolute value on both sides:}
|\vec{r_1} - \vec{r_2}| = |(\vec{v_2} - \vec{v_1})|t
\text{As the collision will happen in future, } t > 0 \text{ and therefore,}
\text{we can take out t from right hand side of the equation:}
|\vec{r_1} - \vec{r_2}| = |\vec{v_2} - \vec{v_1}|t \text{ ...(2)}
\text{Now, dividing (1) by (2) we can get rid of t as below:}
\frac{\vec{r_1} - \vec{r_2}}{|\vec{r_1} - \vec{r_2}|} = \frac{(\vec{v_2} - \vec{v_1})t}{|\vec{v_2} - \vec{v_1}|t}
\text{Simplifying:}
\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}|}
\text{Hence, option (4) is the correct answer.}