here we have to apply to concepts

first is conservation of angular momentam and second is conservation of linear momentam...

initial angular momentam of particle about the center of rod is mva/4 ....

finally particle is rest but rod is rotating with angular velocity W..

so final angular momentam=Iw=initial angular momentam              (by conservation of angular momentam)

            IW=mva/4

 I for rod is ma²/12 about its axis passing through center perpendicular to rod...

             WMa²/12 =mva/4

              W= 3mv/Ma ]   .......................1

now applying conservation of linear momentam

 let V_com is the linear velocity of rod finally then

  MV_com = mv

    V_com  =mv/M .......................2

Approved