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Chapter 35: Problem 35

In your physics class you have just learned about the relativistic frequency shift, and you decide to amaze your friends at a party. You tell them that once you drove through a stop light and that when you were pulled over you did not get ticketed because you explained to the police officer that the relativistic Doppler shift made the red light of wavelength \(650 \mathrm{nm}\) appear green to you, with a wavelength of \(520 \mathrm{nm}\). If your story had been true, how fast would you have been traveling?

Short Answer

Expert verified
Answer: The car would have to travel at a speed of \(1.5 \cdot 10^8 \mathrm{m/s}\) for the red light to appear as the green light.

Step by step solution

01

Write the formula for the relativistic Doppler effect

Using the given wavelengths, write the formula for the relativistic Doppler effect: $$\frac{520 \mathrm{nm}}{650 \mathrm{nm}} = \sqrt{\frac{1 + \beta}{1 - \beta}}.$$
02

Solve for \(\beta\)

Simplify and solve the equation for \(\beta\): $$\frac{520}{650} = \sqrt{\frac{1 + \beta}{1 - \beta}} \Rightarrow \left(\frac{520}{650}\right)^2 = \frac{1 + \beta}{1 - \beta}.$$ Now, set the equation to isolate \(\beta\): $$\beta (1+\beta) = 1 - \beta^2 \Rightarrow \beta^2 + \beta = 1 - \beta^2 \Rightarrow 2\beta^2 + \beta -1 = 0.$$
03

Solve the quadratic equation for \(\beta\)

Use the quadratic formula, \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), to solve for \(\beta\): $$\beta = \frac{-1 \pm \sqrt{(-1)^2 - 4(2)(-1)}}{4} = \frac{-1 \pm \sqrt{9}}{4}.$$ There are two possible solutions for \(\beta\): 1. \(\beta = \frac{-1 + 3}{4} = \frac{1}{2}\) 2. \(\beta = \frac{-1 - 3}{4} = -1\) Since the second solution, \(\beta = -1\), would mean that the car is travelling at the speed of light (which is impossible), we will use the first solution, \(\beta = \frac{1}{2}\).
04

Calculate the speed of the car

Using \(\beta = \frac{v}{c}\), calculate the speed of the car: $$v = \beta \cdot c = \frac{1}{2} \cdot 3 \cdot 10^8 \mathrm{m/s} = 1.5 \cdot 10^8 \mathrm{m/s}.$$ The car would have to travel at a speed of \(1.5 \cdot 10^8 \mathrm{m/s}\) for the story to be true.

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