"A car has acceleration given by a(t)=A sin(ωt) . Find the position x(t)."The unwary student may recognize the form of a(t) as that for simple harmonic motion, and assume the trajectory of the car will be that of simple harmonic motion: back and forth.
But we should recognize that in order to solve for x(t), two initial conditions must be specified: x(0) and v(0). Here we assume x(0)=0, but we experiment with various values of v(0)=v0.
Consider v0=0. Such a value would be appropriate for a car starting from rest, with a traffic light changing to green at t=0. In this case the motion is not back and forth. The car does proceed down the road, perhaps acceleration and decelerating at a series of idealized traffic lights.