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The yellow caution light on a traffic signal should stay on long enough to allow a driver to either pass through the intersection or safely stop before reaching the intersection. A car can stop if its distance from the intersection is greater than the stopping distance fropping distance from the intersection, the yellow light should stay on long enough to allow the car to pass entirely through the inter- section. (a) Show that the yellow light should stay on for a time interval $$ \Delta t_{\text { light }}=\Delta t_{r}-\left(v_{0} / 2 a\right)+\left(s_{i} / v_{0}\right) $$ where \(\Delta t_{r}\) is the driver's reaction time, \(v_{0}\) is the velocity of the car approaching the light at the speed limit, \(a\) is the braking acceleration, and \(s_{i}\) is the width of the intersection. (b) As city traffic planner, you expect cars to approach an intersection 16.0 \(\mathrm{m}\) wide with a speed of 60.0 \(\mathrm{km} / \mathrm{h}\) . Be cautious and assume a reaction time of 1.10 \(\mathrm{s}\) to allow for a driver's indecision. Find the length of time the yellow light should remain on. Use a braking acceleration of \(-2.00 \mathrm{m} / \mathrm{s}^{2}\) .

Step by step solution

01

Analyze the problem

Understand that the total time the yellow light should stay on, \(\Delta t_{\text{light}}\), must accommodate for the driver's reaction time, the time to stop, and the time to pass through the intersection. Follow the given formula to express the required time.

02

Define the variables for part (a)

Let \(\Delta t_{r}\) be the driver's reaction time, \(v_{0}\) the velocity of the car, \(a\) the braking acceleration, and \(s_{i}\) the width of the intersection.

03

Calculate the stopping time

The stopping time \(\Delta t_{s}\) can be found using the equation for constant acceleration: \(v = v_{0} + a\Delta t\). For stopping completely, \(v = 0\), so solving for \(\Delta t_{s}\) gives \(\Delta t_{s} = -\frac{v_{0}}{a}\).

04

Calculate the time to pass through the intersection

The crossing time \(\Delta t_{c}\) is found by dividing the width of the intersection by the car's velocity: \(\Delta t_{c} = \frac{s_{i}}{v_{0}}\).

05

Compose the formula for \(\Delta t_{\text{light}}\)

Combine the driver's reaction time, the time to stop, and the time to pass through according to the given formula for \(\Delta t_{\text{light}}\) to get \(\Delta t_{\text{light}} = \Delta t_{r} + \Delta t_{s} + \Delta t_{c}\).

06

Substitute values for part (b)

Use the given values \(v_{0} = 60.0 \frac{km}{h} = 16.67\frac{m}{s}\) (converted from km/h to m/s), \(a = -2.00\frac{m}{s^2}\), \(\Delta t_{r} = 1.10\ s\), and \(s_{i} = 16.0\ m\).

07

Perform the calculations for part (b)

Substitute the numeric values into the formula and calculate the time interval for the yellow light: \(\Delta t_{\text{light}} = 1.10\ s - \left(\frac{16.67\ m/s}{2 \times -2.00\ m/s^2}\right) + \left(\frac{16.0\ m}{16.67\ m/s}\right)\).

08

Simplify the calculations

Carry out the arithmetic to find \(\Delta t_{\text{light}} = 1.10\ s - (-4.17\ s) + (0.96\ s) = 1.10\ s + 4.17\ s + 0.96\ s = 6.23\ s\).

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Key Concepts

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

Reaction Time

Understanding reaction time plays a pivotal role in traffic signal safety calculations. It's defined as the duration a driver takes to perceive a stimulus, like the turning of a traffic light to yellow, and initiate a response, such as applying the brakes. It sounds straightforward, but human reaction time is influenced by numerous factors, including age, alertness, and even the complexity of the decision-making process.

In our textbook problem, the precautionary reaction time is fairly standardized at 1.10 seconds. This allows for variables such as indecision and ensures there's enough buffer for a range of drivers. It highlights the importance of accommodating human factors when planning traffic systems for safety. To minimize risks at intersections, adequate time must be calculated into the yellow light phase to account for this reaction latency.

Braking Acceleration

Braking acceleration refers to the rate at which a vehicle decreases its speed. Often misinterpreted as deceleration, it's expressed in meters per second squared (\( m/s^2 \)), denoting how quickly velocity is reduced each second. For traffic safety calculations, we consider braking acceleration as a negative value because it signifies a reduction in speed.

For our calculations, we're using a braking acceleration of \( -2.00 m/s^2 \), which implies a relatively gentle stop. Variability in braking acceleration can be due to different vehicle capabilities, road conditions, and even driver behavior. Inertia also plays a role, as heavier vehicles typically experience lower braking accelerations. Therefore, accounting for a conservative braking acceleration, as in this problem, is essential for safety measures in traffic signal timing.

Intersection Crossing Time

When it comes to traffic signal safety, understanding intersection crossing time is indispensable—specifically, how long it takes for a vehicle traveling at a certain speed to traverse the intersection's width. For instance, in our exercise, the width of the intersection, or \( s_i \) is 16.0 meters. Crossed with the speed limit, we determine the minimum time needed to pass through without hastily stopping, which could result in accidents.

It's crucial to recognize that intersection crossing time isn't just dependent on distance and speed; factors like intersection design and driver distraction can also impact the time. However, in the interest of calculations, a simple division of intersection width by the car's velocity (\( v_0 \)) gives a robust estimate. By calculating a proper duration for the yellow light, we merge these concepts to ensure that it serves its purpose: to prevent accidents and promote smooth traffic flow.