Answer to Question #91504 in Mechanics | Relativity for Mis

Question

How much distance (in m) will it take a car to stop when a light turns yellow if it is travelling at the speed limit of 60 km/h? How long should the yellow light be if it is to turn red right after they stop inside this distance?
Drivers do not react right away when they see the light change; human reaction times vary from 0.50 s to 1.0 s. In your calculation, assume the driver does not react for 1.0 s. In other words, the car continues to travel at the speed limit for 1.0 s after the light has turned. Then the driver brakes to a stop at an acceleration of 3.0 m/s2.

2. Using your result from Question 1, determine how long the yellow light needs to be to allow vehicles that are just inside that distance to continue at the speed limit and clear the intersection before the light turns red. For a car to clear the intersection, its back end must be out of the intersection.

Answer 1

Simply solve the problem step-by-step.

1) The driver sees yellow and thinks "Wow, I'd rather slow down now". This thought lasts for 1 second before the driver begins pushing the brake pedal down. Meanwhile the car travels the thinking distance with 60 km/h:

$$d_1=vt_1=60\cdot1000/3600\cdot1=16.67\text{ m}.$$

Then the car begins deceleration to the full stop and the distance required to stop is

$$d_2=\frac{v^2}{2a}=\frac{(60\cdot1000/3600)^2}{2\cdot3}=46.30\text{ m}.$$

The total distance (sees the yellow light and the car stops) hence is

$$d=d_1+d_2=62.96\text{ m}.$$

Quite a lot. And straight after all that the red turns on. By that moment the car stops.

Thus, yellow must last for

$$t_Y=\sqrt{\frac{2d}{a}}=6.48\text{ s}.$$

2) Cars inside that distance move with speed of 60 km/h and they are supposed to continue motion. Hence, add the intersection length $L$ to our braking distance. From this condition yellow should be turned on for

$$t_Y=\frac{d+L}{v}.$$

For medium-size 20 meters intersection this gives time of 4.98 seconds and about 6 seconds for larger intersections, which is inside the braking distance calculated in the first part. So 6.5 seconds of yellow will be enough.