Question

In a race, boy A sees another boy B overtaking him at a speed v. If they were running in opposite directions, speed of B as seen by A is
$
  (a){\text{ = v}} \\
  (b){\text{ < v}} \\
  (c){\text{ > v}} \\
  (d){\text{ }} \geqslant {\text{v}} \\
 $

Answer 1

Hint – In this question use the concept of relative speeds, take one direction as positive and thus the speed in that direction will be positive only and the speed in the direction opposite to it will be negative. There will arise two cases in which the first one will be boy A sees another boy B overtaking him at a speed of v and case two will be when the boys are running in opposite directions. Proper collaboration of both the concepts will help getting the answer.
Step by step answer:
Let the speed of the boy A be ${V_A}$.
And the speed of the boy B be ${V_B}$.
Now according to the theory of relativity that if two bodies travel with different speeds in the same direction then the relative speed is the difference of individual speeds.
And if the two bodies travel with different speeds in opposite directions than the relative speed is the sum of individual speeds.
Case – 1: Boy A sees another boy B overtaking him at a speed of v.
So the relative speed or the overtaking speed of boy B is v m/s.
Now boy A sees another boy B overtaking him so the speed of boy B is greater than the boy A therefore the relative speed is
$ \Rightarrow v = {V_B} - {V_A}$
Case – 2: When the boys run in opposite directions.
So the relative speed is the sum of their individual speeds.
Let the relative speed be v’.
$ \Rightarrow v' = {V_A} + {V_B}$
Now as we see that in case – 2 the relative speed is greater than the relative speed of case – 1.
$ \Rightarrow v' > v$
So when they are running in opposite directions, speed of B as seen by A is greater than the overtaking speed of boy B or the relative speed of boy A and boy B.
So this is the required answer.
Hence option (C) is the correct answer.

Note – The concept of relativity is very helpful in solving problems of this kind. We must have come face to face with problems like two trains travelling opposite to each other at certain velocities, at what time they cross each other. In them the concept of relativity comes into play, we simply make one object stationary by giving its speed in exactly the opposite direction to the other object. This forms the basis of relativity.