a. Let `v` be the velocity of the wire (as well as block )at any instant of time `t`.
Motional emf `e=BvL`
Motional current `i=e/r=(BvL)/L`
and magnetic force on the wire
`F_m=iLB=(vB^2L^2)/R`
Net force on the system at this moment will be
`F_("net")=mg-F_m=mg-(vB^2L^2)/R`
`ma=mg-(vB^2L^2)/R`
`a=(vB^2L^2)/(mR)`
Velocity will acquire its terminal value i.e. `v=v_T` when
`F_("net")` or acceleration `a` of the particle becomes zero.
Thus, `0=g-(v_TB^2L^2)/(mR)`
or `v_T=(mgR)/(B^2L^2)`
b. when `v=v_T/2=(mgR)/(2B^2L^2)`
Then from Eqn (i) acceleratin of the block
`a=g-((mgR)/(2B^2L^2))((B^2L^2)/(mR))=g-g/2`
or `a=g/2`