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Chapter 5: Problem 57

You're in traffic court, arguing against a speeding citation. You entered a 210 -m-radius banked turn designed for \(80 \mathrm{km} / \mathrm{h},\) which was also the posted speed limit. The road was icy, yet you stayed in your lane, so you argue that you must have been going at the design speed. But police measurements show there was a frictional coefficient \(\mu=0.15\) between tires and road. Is it possible you were speeding, and if so by how much?

Short Answer

Expert verified
To answer if you could have been speeding, compare the maximum calculated speed with the design speed. The specific values depend on the calculations in the provided steps.

Step by step solution

01

Calculate the designed speed in m/s

First, we convert the design speed from km/h to m/s because the radius is provided in meters. Use the conversion factor \(1 \mathrm{km} / \mathrm{h} = 0.27778 \mathrm{m}/\mathrm{s}\). So, the design speed \(v_d\) in m/s is calculated as follows: \(v_d = 80 \mathrm{km}/\mathrm{h} \times 0.27778 \mathrm{m}/\mathrm{s}\)
02

Calculate the maximum speed

Next, using the formula for the maximum speed \(v_m\) on a banked road \(v_m = \sqrt{(r \cdot g \cdot tanθ + μr \cdot g) / (1 - μ \cdot tanθ)}\), where \(r\) is the radius of the turn, \(g\) is the acceleration due to gravity, \(θ\) is the banking angle, and \(μ\) is the frictional coefficient. Since the car stayed in its lane, the banking angle \(θ\) is 0. Thus, the equation simplifies to \(v_m = \sqrt{(210 \cdot 9.8 + 0.15 \cdot 210 \cdot 9.8)}\).
03

Compare the maximum speed and the design speed

To determine if you were speeding, compare the maximum speed \(v_m\) calculated in step 2 and the design speed \(v_d\) calculated in step 1. If \(v_m\) is higher than \(v_d\), then you could have been speeding.

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Most popular questions from this chapter

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