The city is trying to figure out how long the traffic light should stay yellow at an intersection. The speed limit on the road is 45.0 km/h and the intersection is 23.0 m wide. A car is traveling at the speed limit in the positive direction and can brake with an acceleration of -5.20 m/s2. (a) If the car is to stop on the white line, before entering the intersection, what is the minimum distance from the line at which the driver must apply the brakes? (3sigfigs)

Vo = 45km/h = 45000m/3600s = 12.5 m/s.

V^2 = Vo^2 + 2a*d.
0 = (12.5)^2 - 2*5.2*d.
d = ?.

To find the minimum distance from the white line at which the driver must apply the brakes, we need to calculate the stopping distance of the car.

First, we'll determine the time it takes for the car to stop. To do this, we'll use the equation of motion:

v = u + at

where:
v = final velocity (0 m/s since the car comes to a stop)
u = initial velocity (45.0 km/h converted to m/s)
a = acceleration (-5.20 m/s²)

Converting the speed limit from km/h to m/s:
45.0 km/h = (45.0 km/h) * (1000 m/km) * (1 h/3600 s) = 12.5 m/s (rounded to 3 decimal places)

Now, substituting the known values into the equation:

0 = 12.5 m/s + (-5.20 m/s²) * t

Solving for t:

5.20 m/s² * t = 12.5 m/s
t ≈ 12.5 m/s / 5.20 m/s²
t ≈ 2.404 s (rounded to 3 decimal places)

The time it takes for the car to stop is approximately 2.404 seconds.

Next, let's calculate the distance traveled during the braking period by using the formula:

s = ut + (1/2)at²

where:
s = distance traveled
u = initial velocity (45.0 km/h converted to m/s)
t = time taken to stop (2.404 s)
a = acceleration (-5.20 m/s²)

Substituting the known values:

s = (12.5 m/s * 2.404 s) + (1/2)(-5.20 m/s²)(2.404 s)²

Calculating:

s ≈ (30.100 m) + (1/2)(-5.20 m/s²)(5.777616 s²)
s ≈ 30.100 m + (-66.736 m)
s ≈ -36.636 m

The distance traveled during braking is approximately -36.636 meters. Since distance cannot be negative, we take its magnitude:

|s| = |-36.636 m| = 36.636 m

Therefore, the minimum distance from the white line at which the driver must apply the brakes is approximately 36.636 meters.