It is possible to start seeing math problems everywhere. For example, you are stopped at a traffic light, several cars from the front. When the light turns green, you notice that the lead car from the other direction coming your way. You know from experience that if you are very far back from the light, the car coming the other way will pass you before you even let your foot off the brake. But if you are closer to the light, you will begin to move before the lead car from the other direction passes you. We will try to set up an equation representing whether you are one of the cars that is moving or one that is still stopped when the lead car from the other direction passes you.

First our assumptions. The lead cars begin a distance x apart. For simplicity, we will assume that the distance between the front of one car to the next is always l, and the reaction time for a car to begin moving after it sees that it is safe (the car in front of it has begun moving, or for the lead cars, the light just changed green) is Dt.

1) Now assuming the lead car reaches the speed v instantaneously, find an equation for n in terms of x, l, v, and Dt that must be satisfied for the n^th car to be moving before the lead car from the other direction reaches its front bumper.

2) Find the number of cars the lead car will pass before it reaches a car that has not yet started to move if x = 16 meters, l = 3 meters, Dt = 0.4 seconds, and v = 16 m/s.
To clarify: Both lead cars begin 0.4 seconds after the light turns green. Car number two begins 0.8 seconds after the light turns green (0.4 seconds after car number one began moving), and so on.

3) If instead we have a constant acceleration a up to the speed v, find an equation for n in terms of x, l, v, a, and Dt that must be satisfied for the n^th car to be moving before the lead car from the other direction reaches its front bumper.

4) Find the number of cars the lead car will pass before it reaches a car that has not yet started to move if x = 16 meters, l = 3 meters, Dt = 0.4 seconds, v = 16 m/s, and a = 5 m/s per second.