Question

Assertion: Relative velocity of A w.r.t B is greater than the velocity of either, when they are moving in opposite directions.
Reason: Relative velocity of A w.r.t B = \[\overrightarrow {{v_A}} - \overrightarrow {{v_B}} \]
A) Both assertion and reason are correct and the reason is the correct explanation for the assertion
B) Both assertion and reason are correct and the reason is not the correct explanation for the assertion
C) Assertion is correct but reason is incorrect.
D) Both assertion and reason are incorrect.

Answer 1

Hint:
According to the question, velocities are in the opposite direction. First, determine the velocity of A w.r.t B by applying the given condition. Then, relate the result with the assertion as well as the reason to find out the correctness of both statements. Further, choose the option which matches the valid outcome.



Complete step by step solution:
Let us assume that there are two bodies A and B respectively and the velocity of the A is $\overrightarrow {{v_A}}$ and the velocity of the B is $\overrightarrow {{v_B}}$. And the velocity of the A w.r.t the velocity of B is $\overrightarrow {{v_{AB}}}$.
Now according to the question, both the bodies A and B are moving in the opposite direction. Suppose that body B is moving in the opposite direction to body A. Therefore, the velocity of body B will be $ - \overrightarrow {{v_B}}$.
Now we know that the relative velocity of the body with respect to another body may be determined as,
$\overrightarrow{v_{12}}=\overrightarrow{v_{1}}-\overrightarrow{v_{2}}$
Therefore,
$\Rightarrow\overrightarrow{v_{AB}}=\overrightarrow{v_{A}}-\lgroup~-\overrightarrow{v_{B}}\rgroup$
$\Rightarrow\overrightarrow{v_{AB}}=\overrightarrow{v_{A}}+\overrightarrow{v_{B}}$
Now, we know that magnitude of the relative velocity is greater than the velocity of either A or B. Therefore,
$\mid\overrightarrow{v_{AB}}\mid\geqslant\mid\overrightarrow{v_{A}}+\overrightarrow{v_{B}}\mid$
Therefore, from the above relation, we can conclude that the assertion is true and the reason is incorrect.
Also, if both bodies A and B are moving in the same direction. Then, the velocity of A w.r.t B will be $\overrightarrow {{v_A}} - \overrightarrow {{v_B}}$. Thus, for the condition mentioned in the assertion, the reason is incorrect.

The correct option is C.





Note:
In this question, the first point is to keep in mind when the two bodies are moving in the opposite direction to each other, then the velocity of either A or B may be taken as negative. In other words, we can say that we should use the sign convention properly.