Average velocity is defined as the ratio in change in position to change in time,
v[ave] = ∆x/∆t
which on its own doesn't have anything to do with acceleration.
<u>If acceleration is constant</u>, the average velocity is the literal average of the initial and final velocities,
v[ave] = (v[final] + v[initial]) / 2
If this constant acceleration has magnitude a, the final velocity can be expressed in terms of the initial velocity by
v[final] = v[initial] + a*t
and plugging this into the previous equation gives
v[ave] = (v[initial] + a*t + v[initial])/2
v[ave] = v[initial] + 1/2*a*t
If the body in consideration is <u>initially at rest</u>, then
v[ave] = 1/2*a*t
which might be the relation you're looking for. But bear in mind the conditions I've underlined.
<u>If acceleration is not constant and changes over time</u>, so that the acceleration is some function of time a(t), then you can determine the velocity function v(t) by using the fundamental theorem of calculus. You need to know a particular velocity for some time to completely characterize v(t), though. For example, if you're given the initial velocity v[initial] = v(0), then
or if you know any other velocity for some time t₀ > 0,