If the displacement of an object is proportional to the square of the time taken, the body is moving with a uniform acceleration because it will follow Newton’s second equation of motion for a particular initial velocity. From second equation of motion,
s = ut + 1/2 at²
If object starts from rest i.e., its initial velocity (u) = 0 and has an acceleration (a) in time (t),
Then,
s = 0 + 1/2 at²
s = 1/2 at²
Now,
s ∝ t², if a = constant
So, the object moves with constant or uniform acceleration.