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Chapter 5: Problem 11

The number of seconds \(n\) for the yellow light is critical to safety at a traffic signal. One study recommends a formula for setting the time that permits a driver who sees the yellow light shortly before entering the intersection either to stop the vehicle safely or to cross the intersection at the current approach speed before the end of the yellow light. \({ }^{69}\) For a street of width 70 feet under standard conditions, the formula is $$ n=1+\frac{v}{30}+\frac{90}{v} . $$ Here \(v\) is the approach speed in feet per second. (See Figure 5.127.) a. Make a graph of \(n\) as a function of \(v\). Include speeds from 30 to 80 feet per second (roughly 20 to 55 miles per hour). b. Express using functional notation the length of the yellow light when the approach speed is 45 feet per second, and then calculate that value. c. Explain in practical terms the behavior of the graph near the pole at \(v=0\). d. What is the minimum length of time for a yellow light?

Short Answer

Expert verified
a) Graph the function for 30-80 fps. b) \( n(45) = 4.5 \) sec. c) As \( v \to 0 \), \( n(v) \to \infty \). d) Minimum time \( \approx 4.08 \) sec.

Step by step solution

01

Understand the Formula

The formula given is \( n = 1 + \frac{v}{30} + \frac{90}{v} \), where \( v \) is the speed in feet per second. Our task is to analyze this function, graph it, calculate specific values, and understand its behavior.
02

Graphing the Function

To graph \( n(v) = 1 + \frac{v}{30} + \frac{90}{v} \) for speeds 30 to 80 feet per second, we calculate \( n \) values at several points, using \( v = 30, 40, 50, 60, 70, 80 \). Choose a graphing tool to plot these points and draw the curve, ensuring to mark axes appropriately.
03

Evaluate Function at Specific Speed

To find the yellow light duration at 45 feet per second, substitute \( v = 45 \) into the formula: \[ n(45) = 1 + \frac{45}{30} + \frac{90}{45} = 1 + 1.5 + 2 = 4.5 \].
04

Behavior Near \( v = 0 \)

As \( v \) approaches 0, the term \( \frac{90}{v} \) becomes very large, causing \( n(v) \) to approach infinity. Practically, this means if a vehicle's speed is very low, the yellow light duration tends to become infinitely long, which aligns with the need for more time to cross safely.
05

Minimum Yellow Light Duration

To find the minimum \( n \), we take the derivative to find critical points: \[ n'(v) = \frac{1}{30} - \frac{90}{v^2} \]. Set \( n'(v) = 0 \) to solve for \( v \): \( \frac{1}{30} = \frac{90}{v^2} \), resulting in \( v^2 = 2700 \), thus \( v = \sqrt{2700} \approx 52 \). Evaluate \( n(52) \) to find minimum time: \[ n(52) = 1 + \frac{52}{30} + \frac{90}{52} \approx 4.08 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Yellow Light Duration
Understanding the duration of a yellow light is crucial for traffic safety. The yellow light signifies to a driver that they should prepare to stop or safely pass through an intersection before the light turns red. The study suggests that the time given for a yellow light, represented as \( n \), is essential for a driver traveling at a certain speed \( v \). This ensures safety, providing enough stopping or crossing time. The yellow light duration formula \( n = 1 + \frac{v}{30} + \frac{90}{v} \) is designed to balance these needs.
The formula captures two main concepts: while faster speeds require slightly longer reaction and crossing times (hence the \( \frac{v}{30} \) component), slower speeds need even more time due to extended stopping distances, represented by \( \frac{90}{v} \).
Thus, understanding this formula helps in setting yellow light durations that match the conditions of various driving speeds, ensuring driver and pedestrian safety alike.
Traffic Signal Timing Formula
The formula \( n = 1 + \frac{v}{30} + \frac{90}{v} \) is pivotal for determining the length of a yellow light based on a driver's approach speed in feet per second. Let's break it down.
  • The term \( 1 \) represents the base second that accounts for a general reaction time.
  • \( \frac{v}{30} \) is directly proportional to the vehicle's speed; the faster you go, the greater this component becomes, acknowledging that higher speeds require more crossing time.
  • \( \frac{90}{v} \) is inversely proportional to speed, reflecting the need for more stopping time as speed decreases, which effectively prevents potential crashes due to abrupt stops at lower speeds.
These terms work together to adjust the yellow light duration to accommodate varying speeds, encouraging safe driving practices and minimizing intersection accidents. Traffic engineers rely on this formula to enhance road safety dynamically.
Function Graphing
Graphing the function \( n(v) = 1 + \frac{v}{30} + \frac{90}{v} \) offers a visual understanding of how yellow light duration changes with speed. When plotting this graph for speeds from 30 to 80 feet per second, expect to see a curve.
Why is graphing valuable here? It allows you to identify trends:
  • The graph typically decreases and then slightly rises as \( v \) increases from lower values.
  • Notice how steep the curve is when \( v \) is small, showing the dramatic impact of speed on yellow light duration at lower speeds.
  • As \( v \) becomes larger, the increase in duration becomes more gradual, illustrating minor adjustments needed at higher speeds.
This graph helps students and engineers alike appreciate how mathematical functions translate to practical tools in traffic management.
Critical Points Calculation
Critical points in the function are special values of \( v \) where the yellow light duration \( n(v) \) is minimized or maximized. To identify critical points, we assess the derivative of the function \( n'(v) = \frac{1}{30} - \frac{90}{v^2} \).
Setting the derivative to zero helps find any speeds where changes in duration are minimized or maximized:
\[ \frac{1}{30} = \frac{90}{v^2} \]
Solving this gives \( v^2 = 2700 \), meaning \( v \approx 52 \) feet per second represents a critical point.
At this speed, we find the yellow light duration by substituting back into the original formula:
\[ n(52) = 1 + \frac{52}{30} + \frac{90}{52} \approx 4.08 \]\( v = 52 \) feet per second is where the yellow light duration reaches its minimum, allowing engineers to optimize light timing to prevent traffic congestion and enhance safety.

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