Work done in time t on a body of mass m accelerated from rest to a speed

Work done in time t on a body of mass m accelerated from rest to a speed v in time t1 as function of time t, is?

Work done = image
where F : force
image
: displacement.
Now according to the problem:
initial velocity = 0
velocity after time t1 = v
therefore acceleration is
image
therefore force F = ma
=image
displacement after time t is
image
hence work done =
image