Consider this traffic problem that is played out in many communities everyday! A car is cruising down the highway of life @ 25m/s in a 3-m long compact car when he comes upon an intersection with a stop sign. At the instant that he is 45 m away from the entrance to the intersection he spots a 4-m long limousine traveling @ 10m/s approaching the intersection from the right perpendicular to him and they are 15m from the entrance to the intersection with NO STOP SIGN Now comes the dilemma: should he "step on it" and accelerate at +5m/s/s or should he brake and decelerate @ -6m/s/s n order to avoid a collision? The intersection is 20m wide in both directions and the cars reaction time is 0.6s.
To determine whether the driver should "step on it" and accelerate or brake to avoid a collision, we need to analyze the situation and calculate the distances each car will travel during the reaction time and potential braking/acceleration.
1. Calculate the distance traveled during the driver's reaction time:
During the driver's reaction time, both cars will continue to move. The distance traveled by the car at 25 m/s during a reaction time of 0.6s is:
Distance_1 = speed * reaction time
Distance_1 = 25 m/s * 0.6 s = 15 m
2. Calculate the distance the car will travel if the driver accelerates:
If the driver chooses to accelerate at 5 m/s^2, the distance traveled by the car during the reaction time plus the distance required to stop can be calculated. We'll refer to this as "Distance_Accel".
Distance_Accel = Distance_1 + (0.5 * acceleration * time^2)
Distance_Accel = 15 m + (0.5 * 5 m/s^2 * (0.6s)^2)
Distance_Accel = 15 m + 0.9 m = 15.9 m
3. Calculate the distance the limousine will travel during the driver's reaction time:
During the driver's reaction time, the limousine traveling at 10 m/s will cover the following distance:
Distance_Limo = speed * reaction time
Distance_Limo = 10 m/s * 0.6 s = 6 m
4. Determine the remaining distance between the cars after the reaction time:
The initial distance between the cars was 45 m, and both cars traveled some distance during the reaction time. Therefore, the remaining distance between the two cars is found by subtracting the distance traveled by the limousine and the car during the reaction time from the initial distance.
Remaining distance = initial distance - Distance_Limo - Distance_1
Remaining distance = 45 m - 6 m - 15 m = 24 m
5. Calculate the time it will take for the car to either stop or collide:
Next, we need to calculate the time it will take for the car to either stop or collide with the limousine in case of acceleration or deceleration. We'll refer to this as "Time_Accel".
Time_Accel = remaining distance / (speed + acceleration * time)
Time_Accel = 24 m / (25 m/s + 5 m/s^2 * time)
6. Calculate the distance the car will travel if the driver decelerates:
If the driver chooses to decelerate at -6 m/s^2, the distance traveled by the car during the reaction time plus the distance required to stop can be calculated. We'll refer to this as "Distance_Decel".
Distance_Decel = Distance_1 + (speed * time) + (0.5 * acceleration * time^2)
Distance_Decel = 15 m + (25 m/s * time) + (0.5 * -6m/s^2 * time^2)
7. Determine the remaining distance between the cars after the reaction time:
As in step 4, the remaining distance between the cars is determined by subtracting the distance traveled by the limousine and the car during the reaction time from the initial distance.
Remaining distance = initial distance - Distance_Limo - Distance_1
Remaining distance = 45 m - 6 m - 15 m = 24 m
8. Calculate the time it will take for the car to either stop or collide:
Similarly, we need to calculate the time it will take for the car to either stop or collide with the limousine in case of deceleration. We'll refer to this as "Time_Decel".
Time_Decel = remaining distance / (speed + acceleration * time)
Time_Decel = 24 m / (25 m/s + (-6 m/s^2) * time)
By comparing the time it will take for the car to either stop or collide in case of acceleration or deceleration, we can determine the appropriate action for the driver. The driver should choose the option that results in a smaller time value, as this indicates the possibility of collision occurring earlier.