I. E. Irodov Solution 1.5

Solution to I. E. Irodov in General Physics and H. C. Verma in Concept of Physics is like a bible for student who are appearing for IIT-JEE, JEE/Main and JEE/Advance, UG NEET, AIIMS or any other Engineering and Medical entrance examination. All the questions in these books are of high level, which requires all basics to applied concept of physics. We here makes your task very easy, we have presented complete solution with detailed explanation step by step. The solution of these books teaches students also teachers in a suitable manner and then tests you with some tricky questions. To answer these questions you need to have thorough understanding of the concepts and this is where most students falter.

Problem: 1.5
Two particles, $1$ and $2$, moves with constant velocities ${V_1}$ and ${V_2}$. At the initial moment their radius vectors are equal to ${r_1}$ and ${r_2}$. How must these four vectors be interrelated for the particles to collide?

Solution: 1.5
The displacement vector of the first particle as a function of time is ${S_1} = {r_1} + {V_1}t$ and that of the second will be
${S_2} = {r_2} + {V_2}t$
Thus,
${r_1} + {V_1}t = {r_2} + {V_2}t$
Therefore,
${r_1} - {r_2} = \left( {{V_2} - {V_1}} \right)t$
Since $t$ is only a scalar (1 dimensional) while $r$ and $V$ are vector (more than 1 dimensional), for this condition to be true, $\left( {{r_1} - {r_2}} \right)$ must be aligned in the same direction as $\left( {{V_2} - {V_1}} \right)$.
Thus,
$\frac{{\left( {{r_1} - {r_2}} \right)}}{{\left| {{r_1} - {r_2}} \right|}} = \frac{{\left( {{V_2} - {V_1}} \right)}}{{\left| {{V_2} - {V_1}} \right|}}$