To get the smallest possible A, you want the trailing end of the liquid column to pass the device just as it turns off. Also note that the length of the column of liquid stays the same.
At this point, I would usually draw a couple pictures - what it looks like at t=0 and what it looks like at t=C. In this time, as you said, the device moves V2*C. How much does the liquid move? So you should be able to figure out the length of the column, and from that the injection time A.