Okie my version...
Assume the whole set up is in equibrum...
Thus, before 8N force is applied, tension in both ropes are 9.81N.
When 8N force is applied, looking at the FBD of the 2kg mass. We'll have, 8+9.81N acting on the left and 9.81N still acting on the right... Therefore, the resultant force on the 2kg mass is still 8N to the left. (The 9.81N cancel out each other)
Thus overall resultant force acting on the 2kg mass is 8N to the right... We all know that F=ma. Therefore, at 1s the acceration is thus 8/2=4m/s^2. Velocity also 4m/s
After 1s. When 8N is taken away, applying newton's law of motion, assuming zero frictional force. We'll have Acceration=0, velocity=4m/s.
My 2 cent's worth...