The observer(2) will observer a pseudo force of magnitude `ma_2` in the direction opposite to the acceleration of observer(2) as shown in figure.
The displacement of the block will respect to trolley (observer2),
`d_(12)=1/2a_(12)t^2`
Hence, work done by pseudo force on block as shown by observer(2),
`W_2=-(ma_2)*(d_(12))`
`=-ma_21/2a_(12)t^2`
`=-1/2ma_2*a_(12)t^2`
The observer(3) will observe a pseudo force `ma_3(larr)`.
For calculating the work done by pseudo force as seen from observer(3) `(W_3)`, we need to calculate displacement of the block w.r.t. observer (3) (i.e., `s_(13)`).
`vecs_(13)=1/2veca_(13)t^2`
`a_(13)=veca_1-veca_3=(veca_(12)+veca_2)-veca_3=(a_(12)+a_2-a_3)`
`impliess_(13)=1/2(a_(12)+a_2-a_3)t^2`
Hence, `W_3=-(ma_3)s_(13)=-(ma_3)[1/2(a_(12)+a_2-a_3)t^2]`
`veca=veca_(13)=veca_1-veca_3=veca_(12)+veca_2-veca_3=(a_(12)+a_2-a_3)t^2`
Using the above equations, we have
`W_(pseudo)=-(ma_3)/(2)(a_(12)+a_2-a_3)t^2`