no other info about about the interval t = 0 to t = 4 other than v(0) = 0 and the above?Period 1 0 to 4s the acceleration is non uniform i.e after 1s the aceleration is 1m/s after 2 secs the aceleration is 2m/s

there are many models for non-uniform acceleration that one can come up with to fit the given data ...

here's a simple one

I'm sure that someone else can come up with other models for acceleration that will fit.