A person driving her car at 45km/h approaches an intersection just as the traffic light turns yellow. She knew that the yellow light lasts only 2s before turning red,and shi is 28m away from the near side of the intersection.
Should she try to stop,or should she speed up to cross the intersection before the light turns red? The intersection is 15m wide. Her car's maximum deceleration is -5.8m/s2,whereas it can accelerate from 45km/h to 65km/h in 6s. Ignore the length of her car and her reaction time
Vo = 45,000m/3600s = 12.5 m/s.
If she tries to stop:
V^2 = Vo^2 + 2a*d.
0 = 12.5^2 - 2*5.8*d.
11.6d = 156.25, d = 13.5 m. To stop. Therefore, she should stop the car.
Well, if she speeds up to cross the intersection before the light turns red, she might end up feeling "blue" if she gets caught by the traffic police. And trust me, they're not fans of drivers running red lights, especially when it's still yellow.
So, it's probably better for her to stop and wait for the light to turn green. We don't want her causing a catastrophe at the intersection or becoming the "brake" of someone's jokes later on. Safety first!
To determine whether the person should stop or speed up to cross the intersection before the light turns red, we need to compare the distance it takes for the car to stop versus the distance it takes for the car to cross the intersection.
1. Calculate the distance it takes for the car to stop:
The car's maximum deceleration is -5.8 m/s^2.
Using the kinematic equation: v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance traveled.
Since the car is currently at 45 km/h = 12.5 m/s, and it needs to come to a stop, v = 0.
The acceleration is -5.8 m/s^2.
The distance traveled can be calculated as: s = (v^2 - u^2) / (2a) = (0 - 12.5^2) / (2 × -5.8) ≈ 27.15 m
2. Determine the time it takes for the car to cross the intersection:
The car's initial velocity is 45 km/h = 12.5 m/s.
The car's maximum acceleration is (65 km/h - 45 km/h) / 6 s = 3.33 m/s^2 (approximated acceleration to reach 65 km/h in 6 seconds).
To calculate the time, we can use the equation: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Since we need to find the time it takes for the car to cross the intersection, v = 0.
Therefore, the equation simplifies to: t = (v - u) / a = (0 - 12.5) / 3.33 ≈ -3.76 s (approximated time to reach 0 m/s)
Since time cannot be negative, we can conclude that it is not possible for the car to accelerate and cross the intersection before the light turns red. The car should try to stop before reaching the intersection to avoid running a red light.
To determine whether the person should try to stop or speed up to cross the intersection before the light turns red, we need to consider the time it would take for the car to stop or accelerate to cross the intersection.
First, let's convert the car's speed from km/h to m/s. We know the car is driving at 45 km/h, so we can multiply it by (1000/3600) to get the speed in m/s.
45 km/h * (1000/3600) = 12.5 m/s
The person is 28m away from the near side of the intersection when the yellow light turns on. We also know that the intersection is 15m wide. So, the total distance the car needs to travel to cross the intersection is 28m + 15m = 43m.
Now, let's consider the car's maximum deceleration and acceleration values.
Maximum deceleration: -5.8 m/s²
Acceleration from 45 km/h to 65 km/h in 6 seconds: (65 km/h - 45 km/h) / 6s = 3.3 m/s²
Now, let's calculate the time it would take for the car to stop or accelerate to cross the intersection.
1. Time to stop:
Using the formula of motion, s = ut + (1/2)at², where s is the distance, u is the initial speed, t is time, and a is acceleration, we can calculate the time it would take for the car to stop.
s = (0 m/s)t + (1/2)(-5.8 m/s²)t²
43m = (1/2)(-5.8 m/s²)t²
86 m = -5.8 t²
t² = 86m / -5.8 m/s²
t² = -14.8 s²
t ≈ 3.8 seconds
2. Time to accelerate:
Using the formula of motion, s = ut + (1/2)at², where s is the distance, u is the initial speed, t is time, and a is acceleration, we can calculate the time it would take for the car to accelerate from 12.5 m/s to 23 m/s (65 km/h).
s = (12.5 m/s)t + (1/2)(3.3 m/s²)t²
43m = (12.5 m/s)t + (1/2)(3.3 m/s²)t²
t² + 7.58t - 8.46 = 0
Using the quadratic formula, we find that t ≈ 0.68 seconds or t ≈ -7.58 seconds. Since time cannot be negative in this context, we consider t ≈ 0.68 seconds.
Considering the time values calculated, we can determine the best course of action:
- Time to stop: 3.8 seconds
- Time to accelerate: 0.68 seconds
Since the yellow light only lasts 2 seconds, the person does not have enough time to either stop or accelerate safely across the intersection. Therefore, the best course of action would be for the person to try to stop rather than risk speeding up and potentially not being able to cross the intersection safely before the light turns red.