What is the relationship between the displacement of an object and time under uniform acceleration?

When the rate of change of velocity of a body is constant then the body is said to be uniformly accelerated.

In case of uniform motion, we use the newton's equations of motion, and second equation of motion is expressed as:

 $\mathrm{S} = \mathrm{ut} + \frac{1}{2} {\mathrm{at}}^2$

If the body starts moving from rest, then u = 0 

$\mathrm{S} = \frac{1}{2} {\mathrm{at}}^2$

Hence, displacement is proportional to the square of time, if the object is uniformly accelerated.