The motion of a toy train in a horizontal x-y plane is such that $\ddot{x} = -16 \cos 2t$ and $\ddot{y} = -16 \sin 2t$ where x and y are measured in feet and t in seconds. If the toy train starts when t = 0 so that x = 4 ft, y = 0, $\dot{x} = 0$, and $\dot{y} = 8$ ft/s, determine (a) the equation of the path followed by the train, (b) the velocity of the train for t = 0.5 s, and (c) the acceleration of the train for t = 0.5 s.
