A block of mass `m_1` moves with an acceleration `a_(12)` relative to a trolley as shown in figure. The block is being observed by two observers (2) and (3). The observer (2) is at rest with respect to trolley which is moving with acceleration `a_2` while the observer(3) is moving on ground with acceleration `a_3`. What is the work done by the pseudo force as observed by the observers (2) and (3) on the block during time t? Assume zero intial velocities of the bodies and observers.

Explanation

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`