I think I’m making a mistake to reach my goal.

I would like to find an $ x[t]$ function that could describe the following conditions:

1 – The object is in position $ x=151$ (meters) at time $ t=5$ (second).

2 – The object is in position $ x=1215$ (meters) at time $ t=59$ (second).

3 – The object has velocity $ v=0$ (meters/second) at time $ t=5$ (second).

4 – The object has velocity $ v=0$ (meters/second) at time $ t=59$ (second).

5 – The object has acceleration $ a=1$ (meters/second^2) at time $ t=5$ (second).

6 – The object has acceleration $ a=0$ (meters/second^2) at time $ t=59$ (second).

DSolve[{x[5] == 151, x[59] == 1215, x'[5] == 0, x'[59] == 0,    x''[5] == 1, x''[59] == 0}, x[t], t] 

The goal

The idea is that the object be stopped at position $ x=151$ , be accelerated and decelerate at a certain point in the path (which may actually be random), so that when it reaches position $ x=1215$ , it stops.

EDIT

Maybe I have to use Piecewise, because I want 1/3 of the path to be linear.