##
Kinematics of stop-and-go traffic

### What is this?

Consider the following question:
"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)=v_{0}.

Consider v_{0}=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.