Question

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 stoplight and 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 were true, how fast would you have been traveling?

Short answer

Answer: If the story were true, the narrator would have been traveling at a speed of approximately \(1.0976 \times 10^8\, \text{m/s}\).

Step-by-step solution

Write down the given information

We are given the following information: - Initial wavelength of red light, \(\lambda_R = 650\,\text{nm}\) - Perceived wavelength of green light, \(\lambda_G = 520\,\text{nm}\) Our goal is to find the speed at which the narrator was traveling.

Use the formula for the relativistic Doppler shift

The formula for the relativistic Doppler shift is given by: \(\lambda_G = \lambda_R \sqrt{\frac{1+\beta}{1-\beta}}\) where - \(\beta = \frac{v}{c}\) is the ratio of the speed of the observer (\(v\)) to the speed of light (\(c\)) - \(\lambda_G\) is the perceived wavelength (green light) - \(\lambda_R\) is the initial wavelength (red light) We are given \(\lambda_G\) and \(\lambda_R\); we need to find \(\beta\) and then calculate \(v\).

Solve for \(\beta\)

First, isolate \(\beta\) in the above formula: \(\frac{\lambda_G}{\lambda_R} = \sqrt{\frac{1+\beta}{1-\beta}}\) Square both sides of the equation: \(\left(\frac{\lambda_G}{\lambda_R}\right)^2 = \frac{1+\beta}{1-\beta}\) Next, multiply both sides by \((1-\beta)\): \(\left(\frac{\lambda_G}{\lambda_R}\right)^2 (1-\beta)=1+\beta\) Now, expand the left side of the equation: \(\left(\frac{\lambda_G^2}{\lambda_R^2}\right) - \beta\left(\frac{\lambda_G^2}{\lambda_R^2}\right) =1+\beta\) Rearrange the equation to isolate \(\beta\): \(\beta\left(1+\frac{\lambda_G^2}{\lambda_R^2}\right)=\frac{\lambda_G^2}{\lambda_R^2}-1\) Finally, solve for \(\beta\): \(\beta=\frac{\frac{\lambda_G^2}{\lambda_R^2}-1}{1+\frac{\lambda_G^2}{\lambda_R^2}}\)

Calculate \(\beta\)

Substitute the given values of \(\lambda_G\) and \(\lambda_R\) into the formula obtained in step 3: \(\beta=\frac{\frac{(520\,\text{nm})^2}{(650\,\text{nm})^2}-1}{1+\frac{(520\,\text{nm})^2}{(650\,\text{nm})^2}}\) Calculating the value, we get: \(\beta \approx 0.36587\)

Calculate the speed \(v\)

Recall that \(\beta = \frac{v}{c}\). We can find \(v\) by multiplying \(\beta\) by the speed of light (\(c \approx 3 \times 10^8\,\text{m/s}\)): \(v = \beta \cdot c\) \(v \approx 0.36587 \cdot 3 \times 10^8\,\text{m/s}\) Calculating the value, we get: \(v \approx 1.0976 \times 10^8\,\text{m/s}\) So, if the story were true, the narrator would have been traveling at a speed of approximately \(1.0976 \times 10^8\, \text{m/s}\).

Key concepts

Wavelength of Light

The wavelength of light is a fundamental concept in physics that refers to the distance between consecutive crests of a wave. It essentially dictates the color of light that we perceive; each color within the visible spectrum has a specific wavelength range. For example, red light typically has a wavelength of around 650 nanometers (nm), while green light has a wavelength closer to 520 nm.

Understanding the wavelength is crucial when studying optical phenomena, including the relativistic Doppler shift, which occurs when an object emitting light (such as a car's headlights) is moving at a high rate of speed relative to an observer. In the exercise, we explore how a change in perceived wavelength can theoretically occur due to the high-speed movement of a car, shifting the red light into a green light's wavelength range.

Speed of Light

The speed of light, often represented by 'c', is a constant that plays a significant role in many areas of physics, particularly in the realm of special relativity. It is the speed at which all light waves propagate in a vacuum and is approximately 299,792,458 meters per second (or about 3 x 10^8 m/s).

When discussing the relativistic effects on light, such as the Doppler shift, it's important to consider the speed of light because it serves as the cosmic speed limit; nothing can travel faster than light in a vacuum. In the given exercise, we use the speed of light to determine how fast the narrator's car would need to travel to observe a significant shift in the wavelength of light from red to green.

Special Relativity

Special relativity is a theory put forth by Albert Einstein, which revolutionizes our understanding of time, space, mass, and energy. One of its most profound implications is that the laws of physics are the same for all non-accelerating observers and that the speed of light within a vacuum is the same no matter the speed at which an observer travels. This leads to phenomena like time dilation and length contraction, and it fundamentally alters the way we understand motion at speeds close to the speed of light.

The relativistic Doppler shift is a direct application of special relativity, taking into account the fact that as objects move at speeds approaching the speed of light, we must account for these relativistic effects when predicting how the wavelength and frequency of light changes.

Doppler Effect

The Doppler effect refers to the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. It is commonly experienced in everyday life, such as when a siren sounds higher in pitch as it approaches us and lower as it moves away. This effect is not limited to sound; it also applies to light waves.

In the context of light and special relativity, this phenomenon is known as the relativistic Doppler shift. It is essential when considering objects moving at substantial fractions of the speed of light. The exercise exemplifies this concept by demonstrating how the wavelengths of light perceived by the observer would shift due to high-speed motion. For light, moving toward a source compresses (blue-shifts) the waves, while moving away stretches (red-shifts) them, but the added relativistic component gives rise to more complex calculations, especially as velocities approach the speed of light.