91Ó°ÊÓ

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 found in the previous problem. If the car is less than this stopping distance from the intersection, the yellow light should stay on long enough to allow the car to pass entirely through the intersection. (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 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

Convert the velocity from km/h to m/s

The given car's velocity \(v_{0}\) is 60.0 km/h. Convert this from km/h to m/s by multiplying with \(1000/3600 = 5/18\). Therefore, \(v_{0} = 60.0 * 5/18 = 16.67 m/s\)

02

Plug in the given values into the formula of time interval

Now that all variables are known or calculated, we can substitute them into the given time interval formula. Thus: \(\Delta t_{\text {light }} = 1.10s - (16.67 m/s / 2*-2.00 m/s²) + (16.0 m / 16.67 m/s)\)

03

Solve for the time interval

Calculate the time intervals: \(\Delta t_{\text {light }} = 1.10s - (-4.17s) + (0.96s) = 6.23s\)

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Key Concepts

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

Driver Reaction Time

Driver reaction time is the period it takes for a driver to respond to a visual or auditory signal. In the context of traffic signal timing, it is crucial because it determines how quickly a driver can begin to react once they notice a yellow traffic light.
This reaction time typically ranges between 0.75 seconds to 1.5 seconds but may vary based on several factors:

• Alertness of the driver
• Unexpected obstacles or distractions
• Age and experience of the driver

In our example, we use a reaction time of 1.10 seconds to account for the possibility of hesitation or indecision when approaching an intersection. It is essential to factor in this response time to ensure traffic lights stay yellow long enough, allowing drivers to decide whether to stop or proceed safely.

Braking Acceleration

Braking acceleration is the rate at which a vehicle slows down when the brakes are applied. This is an important factor in our traffic signal timing calculation as it impacts the distance required to come to a complete stop when approaching a yellow light.
The formula:

• Braking acceleration is generally negative since it indicates a decrease in speed.
• In this exercise, we assume a braking acceleration of \(-2.00\, \text{m/s}^2\).

This assumes standard braking conditions on a typical road surface. However, real-world factors such as road conditions, tire quality, and weather can affect braking ability. Ensuring the yellow light duration adequately accounts for the braking distance ensures driver safety and traffic efficiency.

Kinematic Equations

Kinematic equations are used to describe the motion of objects. In traffic engineering, they help in calculating the necessary yellow light duration by considering speed, acceleration, and distance.
Our exercise utilizes a kinematic approach through the following equation: \[\Delta t_{\text{light}} = \Delta t_{r} - \left( \frac{v_{0}}{2a} \right) + \left( \frac{s_{i}}{v_{0}} \right)\]where:

• \(\Delta t_{r}\) is the driver's reaction time.
• \(v_{0}\) represents the initial velocity of the car.
• \(a\) is the braking acceleration.
• \(s_{i}\) symbolizes the width of the intersection.

The term \(\frac{v_{0}}{2a}\) calculates the extra time needed to stop, while \(\frac{s_{i}}{v_{0}}\) is the time required to cross the intersection. Kinematic equations are straightforward but require careful variable management to achieve accurate results in traffic light timing.