A particle is observed from two frames \[{S_1}\] and \[{S_2}\]. The frame \[{S_2}\] moves with respect to \[{S_1}\] with an acceleration \[\alpha \]. Let \[{F_1}\]and \[{F_2}\] be the pseudo forces on the particle when seen from \[{S_1}\] and \[{S_2}\] respectively. Which of the following are not possible?
A. \[{F_1} = 0,{F_2} \ne 0\]
B. \[{F_1} \ne 0,{F_2} = 0\]
C. \[{F_1} \ne 0,{F_2} \ne 0\]
D. \[{F_1} = 0,{F_2} = 0\]

Hint:In this question we are given with a particle which is being observed by the frames \[{S_1}\] and \[{S_2}\]. We are also given that the pseudo force \[{F_1}\] and \[{F_2}\] is acting on the particle. We are given the four conditions in which we need to tell weather which condition will not be possible.

Complete step by step solution:
Given the frames \[{S_1}\] and \[{S_2}\], Pseudo forces \[{F_1}\] and \[{F_2}\]. We are given a particle which is being observed by the two frames. Let us first consider that the frame \[{S_2}\] is moving with an acceleration \[\alpha \] with respect to the frame \[{S_1}\], now according to the pseudo force if there is no force acting on frame \[{S_1}\]i.e\[{F_1} = 0\left( {{\alpha _1} + {\alpha _2}} \right)\hat i\] then the frame \[{S_2}\] will move in another direction of particle with a force \[{F_2} \ne 0\].

Now, similarly consider that \[{S_1}\] is moving with an acceleration \[\alpha \] with respect to the frame \[{S_2}\], so the force acting on frame \[{S_2}\] i.e., \[{F_2} = 0\] then the frame \[{S_2}\] will move in another direction of the particle with a force \[{F_1} \ne 0\].

Now, from the above two condition let us consider frame \[{S_1}\] is moving with an acceleration \[{\alpha _1}\] in one direction and frame \[{S_2}\] is moving with an acceleration \[{\alpha _2}\] in another direction. Now, observing the frame \[{S_2}\] with respect to the frame \[{S_1}\], we can write \[\left( {{\alpha _1} + {\alpha _2}} \right)\hat i\] in positive x-direction we can say there is relative acceleration in both the body, we can write \[{F_1} \ne 0,{F_2} \ne 0\].

Now, considering the above cases, we can say if one screen will be in rest then the other screen will always be in motion according to the pseudo forces, so the condition \[{F_1} = 0,{F_2} = 0\] will not be possible.

Therefore, option D is not possible.

Note:Pseudo force is also known as the fictitious force which arises when a frame from which is the reference is acceleration with respect to a non-accelerating frame. Pseudo force is observed in all the objects which are accelerated and there is no physical interaction between the objects.