Forgive me but I'm going to change your notation a bit, so that it's less confusing.
v = velocity,
V_i = initial velocity
V_t = target velocity
T = time to reach target velocity
X = distance traveled to reach target velocity
The equation of motion that you are describing is this:
From this we have . Hence the time to reach the target velocity is
The distance traveled in time T is:
Substituting the value for T we get:
Hope this helps!