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Answer :
Final answer:
The direction sign should be installed at least 201.045 meters in advance of the highway off-ramp, factoring in both the perception-reaction distance and the braking distance needed for vehicles to safely decelerate from 50 mph to the 25 mph design speed of the ramp.
Explanation:
To calculate the minimum distance a direction sign should be installed in advance of the ramp, we need to consider the two main components: the distance traveled during the perception-reaction time and the braking distance necessary to slow down to the design speed of the ramp. The highway's 85th percentile speed is 50 mph, which is approximately 22.35 m/s (perception-reaction time of 1.5 seconds). During the perception-reaction time, a constant speed of 50 mph is maintained.
Firstly, we calculate the distance traveled during the perception-reaction time:
- Perception-reaction distance = speed imes perception-reaction time = 22.35 m/s imes 1.5 s = 33.525 m.
Next, we calculate the braking distance using the formula:
Braking distance = (initial velocity) extsuperscript{2} / (2 imes deceleration rate imes g), where g is the acceleration due to gravity (9.81 m/s extsuperscript{2}). Given a deceleration rate of 0.3 g, the calculation is as follows:
- Braking distance = (22.35 m/s) extsuperscript{2} / (2 imes 0.3 imes 9.81 m/s extsuperscript{2}) = 167.52 m.
The total minimum distance for the direction sign placement is the sum of the perception-reaction distance and the braking distance:
- Total minimum distance = perception-reaction distance + braking distance = 33.525 m + 167.52 m = 201.045 m.
Therefore, the direction sign should be installed at least 201.045 meters in advance of the highway off-ramp to ensure that vehicles traveling at the 85th percentile speed have adequate time to react and safely decelerate.
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