The displacement of a particle starts from rest is proportional to the square of time, then the particle travels with

If the displacement of an object is proportional to the square of the time taken then the body is moving with uniformly accelerated motion as it will follow Newton's second equation of motion for a particular initial velocity, which can be given by $\mathrm{s}=\mathrm{ut}+\frac{1}{2}{\mathrm{at}}^2$,

as the particle is starting from rest u=0.

So the displacement is given by $\mathrm{s}=\frac{1}{2}{\mathrm{at}}^2$