Question

Two particles are moving with velocities \[{v_1}\] and \[\;{v_2}\] ​.Their relative velocity is the maximum, when the angle between their velocities is:
A. Zero
B. $\dfrac{\pi }{4}$
C. $\dfrac{\pi }{2}$
D. $\pi $

Answer 1

Hint: In the above given question, When two bodies move in opposite directions, their relative velocity is at its highest. As a result, the angle between their velocities must be $\pi $ so that their relative velocity is as high as possible.

Complete step by step answer:
This will be at its highest when two bodies are approaching each other and at its lowest when two bodies are moving apart in the same line. When two bodies approach, the relative velocity is the sum of the two velocities, and it is the highest.The relative velocity of two bodies travelling away from each other is the magnitude difference between their velocities.

Now, coming to the question; in this given problem the velocities of these two particles are given to us as \[{v_1}\] and \[\;{v_2}\] respectively. Now, the relative velocity according to one another is given by;
$w = \sqrt {\left( {v_1^2 + v_2^2 - 2{v_1}{v_2}\cos \alpha } \right)} $
Now, for the maximum relative velocity ${w_{\max }}$ , $\alpha = {180^ \circ }$
Therefore,
\[{w_{\max }} = \sqrt {\left( {v_1^2 + v_2^2 - 2{v_1}{v_2}\cos {{180}^ \circ }} \right)} \]
Proceeding further in the equation
$\Rightarrow {w_{\max }} = \sqrt {{{\left( {{v_1} + {v_2}} \right)}^2}} \\
\therefore {w_{\max }} = {v_1} + {v_2} $
When the angle between their velocities is \[{180^ \circ }\], the relative velocity is at its maximum.

Hence, the correct option is D.

Note: When two bodies have the same velocity, the distance between them remains constant; however, when their velocity differs, the distance between them grows at a rate that is directly proportional to the velocity difference between them.