Question

How fast would you have to drive in order to see a red light as green? Take \(\lambda=630\) nm for red and \(\lambda=530 \mathrm{nm}\) for green.

Short answer

Answer: The observer would need to drive at a speed of approximately \(3.21 \times 10^7 \mathrm{m/s}\) to perceive the red light as green.

Step-by-step solution

Identify given values and Unknown value

In this exercise, we are given the following values: - Red light wavelength: \(\lambda_{source} = 630 \mathrm{nm}\) - Green light wavelength: \(\lambda_{observed} = 530 \mathrm{nm}\) - Speed of light: \(c = 3 \times 10^8 \mathrm{m/s}\) We are asked to determine the required velocity \(v\) of the observer to perceive the red light as green.

Rearrange the relativistic Doppler effect formula to solve for \(v\)

First, we'll rearrange the Doppler effect formula to solve for the unknown \(v\): $$\frac{\lambda_{observed}}{\lambda_{source}} = \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$$ Square both sides: $$\left(\frac{\lambda_{observed}}{\lambda_{source}}\right)^2 = \frac{1+\frac{v}{c}}{1-\frac{v}{c}}$$ Now, we can cross multiply and isolate \(v\): $$v = c\frac{\left(\frac{\lambda_{observed}}{\lambda_{source}}\right)^2 - 1}{\left(\frac{\lambda_{observed}}{\lambda_{source}}\right)^2 + 1}$$

Plug in the given values and calculate the required velocity \(v\)

Now, we'll plug in the given values for \(\lambda_{observed}\), \(\lambda_{source}\), and \(c\): $$v = (3 \times 10^8) \frac{(\frac{530}{630})^2 - 1}{(\frac{530}{630})^2 + 1}$$ Calculate the required velocity: $$v \approx -3.21 \times 10^7 \mathrm{m/s}$$ Notice that the result is negative, which means the velocity should be in the opposite direction. However, the question asks how fast, not in which direction, so we can take the absolute value: $$|v| \approx 3.21 \times 10^7 \mathrm{m/s}$$

Finalize the answer

The observer would need to drive at a speed of approximately \(3.21 \times 10^7 \mathrm{m/s}\) in order to perceive the red light as green.