"a simple question on frames"

Consider the frames s₁ and s₂.

So imagine the initial position of the frames to be as in the diagram below.

======( S1)================================

<o>

=======( S2)===============================

Since both are accelerating, let the final positions of S1 and S2 be as in the diagram below :

================( S1)======================

<o>

==========================( S2)============

Case 1: Particle <o> is stationary.

There is a relative acceleration between the particle <o> and S1. So a (pseudo) force has to exist.

There is a relative acceleration between the particle and S2 also, so (pseudo) force has to exist.

Hence C is possible.

Case 2: Particle <o> is moving with acceleration equal to S1.

Here acceleration w.r.t S₁ is 0 and w.r.t S₂ there is some value of acceleration.

Hence F₁ = 0, and F₂ ≠0.

Case 3: Particle <o> is moving with acceleration equal to S2.

Here acceleration w.r.t S₁ has some value whereas acceleration w.r.t S₂ is 0.

Hence F₁ ≠0 and F₂ = 0.

So A and B are possible.

Hence the correct solution to this question is D

I have a reason why D is not possible.

It is given that the frames move at an acceleration with respect to each other, hence both frames have different accelerations. So the particle cannot exhibit two accelerations at the same time. Hence the relative accelerations w.r.t both frames cannot be zero.

Hence F₁ ≠0 and F₂ ≠0 situation is not possible. Hence solution D is only correct.

Hope you got the concepts right :)

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