You are driving in a car at 15 m/s, approaching an intersection that is 8.0 meters long. When you are 35 m away from the beginning of the intersection, the light turns yellow – it will turn red in 2.0 seconds. You must decide if you will hit the gas or the brakes. By hitting the gas, you can accelerate at 3.0 m/s2; by hitting the brakes, you can decelerate at 4.0 m/s2. Assume 0.20 seconds of reaction/decision time. The law (in most states) says you may not enter the intersection when the light is red. What should you do? Is there anything unrealistic about this problem?
The correct answer and explanation is :
To determine whether to accelerate or brake upon encountering a yellow light under the given conditions, we need to analyze both scenarios:
Braking Scenario:
- Reaction Distance: During the 0.20-second reaction time, the car continues at its initial speed.
- Distance covered: ( d_{\text{reaction}} = v \times t_{\text{reaction}} = 15\, \text{m/s} \times 0.20\, \text{s} = 3\, \text{m} )
- Braking Distance: After the reaction time, the car decelerates at ( 4.0\, \text{m/s}^2 ) from 15 m/s to a stop.
- Using the equation: ( v^2 = u^2 + 2a s ) (where ( v = 0 ), ( u = 15\, \text{m/s} ), ( a = -4.0\, \text{m/s}^2 ))
- Solving for ( s ): ( 0 = (15\, \text{m/s})^2 + 2(-4.0\, \text{m/s}^2) s )
- ( s = \frac{(15\, \text{m/s})^2}{2 \times 4.0\, \text{m/s}^2} = \frac{225}{8} = 28.125\, \text{m} )
- Total Stopping Distance: ( d_{\text{total stop}} = d_{\text{reaction}} + d_{\text{braking}} = 3\, \text{m} + 28.125\, \text{m} = 31.125\, \text{m} )
Since the car is 35 meters from the intersection, and the total stopping distance is approximately 31.125 meters, braking is a viable option.
Accelerating Scenario:
- Time Available to Accelerate: The light remains yellow for 2.0 seconds, minus the 0.20-second reaction time, leaving 1.8 seconds to clear the intersection.
- Acceleration Distance: Using the equation: ( s = ut + \frac{1}{2} a t^2 ) (where ( u = 15\, \text{m/s} ), ( a = 3.0\, \text{m/s}^2 ), ( t = 1.8\, \text{s} ))
- ( s = 15\, \text{m/s} \times 1.8\, \text{s} + \frac{1}{2} \times 3.0\, \text{m/s}^2 \times (1.8\, \text{s})^2 )
- ( s = 27\, \text{m} + 4.86\, \text{m} = 31.86\, \text{m} )
- Total Distance to Clear Intersection: The car needs to travel 35 meters to the intersection plus 8 meters to clear it, totaling 43 meters.
Since the car can only cover approximately 31.86 meters in the available time, accelerating would not allow it to clear the intersection before the light turns red.
Conclusion:
Given these calculations, braking is the safer and legally compliant choice.
Unrealistic Aspects of the Problem:
- Reaction Time: A reaction time of 0.20 seconds is shorter than the average human reaction time, which typically ranges from 0.25 to 1 second.
- Acceleration and Deceleration Rates: The provided acceleration (3.0 m/s²) and deceleration (4.0 m/s²) rates may not accurately reflect real-world vehicle capabilities, which can vary based on vehicle type and conditions.
- Yellow Light Duration: A yellow light duration of 2.0 seconds is shorter than the standard in many regions, where it often ranges from 3 to 6 seconds to allow drivers adequate time to react.
- Ignoring Vehicle Length: The problem does not account for the car’s length, which affects the distance required to fully clear the intersection.
These factors suggest that the scenario may not fully align with typical driving conditions and standards.