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 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

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

Write the formula for the relativistic Doppler effect

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

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 \(\beta\): $$\beta (1+\beta) = 1 - \beta^2 \Rightarrow \beta^2 + \beta = 1 - \beta^2 \Rightarrow 2\beta^2 + \beta -1 = 0.$$

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, \(\beta = \frac{1}{2}\).

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.

Key concepts

Quadratic Formula in Physics

The quadratic formula is a powerful tool in mathematics and physics, especially when dealing with equations of the form \(ax^2 + bx + c = 0\). Here, it helps us solve problems involving terms that may arise in physics equations, such as relativistic motion. By using the quadratic formula, we can find solutions for variables like \( \beta \), a representation of velocity in terms of the speed of light. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Let’s break it down:

• \(x\) represents the variable we’re solving for.
• \(a\), \(b\), and \(c\) are coefficients from the equation.
• The formula calculates solutions by handling both the positive and negative square roots.

In physics, using this formula helps us in solving for velocities or other quantities where the relationships are quadratic. For the Doppler effect scenario, solving this equation allowed us to find two solutions for \( \beta \), highlighting which speed adheres to physical constraints.

Frequency Shift

The frequency shift is a core component of the relativistic Doppler effect. It describes how the frequency and wavelength of light change when the source and observer are moving relative to each other. This shift is crucial in various fields such as astrophysics and radar technology. To understand it better, imagine you're approaching a source of light. The light waves get compressed due to the motion, causing a shift toward the blue end of the spectrum (blue shift). Conversely, when moving away, the waves stretch, shifting the frequency toward the red end (red shift). In our exercise, the red stop light appeared green, meaning the light shifted significantly due to the relative high speed of the observer. The relationship between observed wavelength \( \lambda' \) and original wavelength \( \lambda \) can be expressed with the formula: \[ \frac{\lambda'}{\lambda} = \sqrt{\frac{1 + \beta}{1 - \beta}} \] Here, \( \beta \) symbolizes the object's speed relative to the speed of light, dictating the degree of frequency shift. Understanding this shift is key to analyzing motion in relativistic contexts.

Speed of Light

The speed of light (\(c\)) is a constant and fundamental limit in physics, with a value of approximately \(3 \times 10^8\) meters per second. It's not just a massive number; it serves as a cornerstone of Einstein's theory of relativity. No object with mass can reach or exceed this speed. In the realm of relativity, the speed of light is the ultimate speed limit. It plays a critical role in how we measure time, distance, and speed in physics. For the Doppler effect, it's used to calculate the velocity of moving objects in relation to the speed of light. The term \( \beta \) is used to simplify calculations, defined as: \[ \beta = \frac{v}{c} \] This relates an object's speed \(v\) to the speed of light. Whenever we consider phenomena at near-light speeds, the behavior of light and objects changes dramatically, requiring the variables in our formulas to reflect this natural constant. In our exercise, we used \( \beta \) to calculate what speed would be needed to experience such a significant frequency shift.

Relativity in Motion

Relativity in motion centers around how time, space, and velocity impact observers differently, particularly when traveling near the speed of light. This concept shifts our understanding of conventional motion to one that accounts for the effects proposed by Einstein's theory of relativity.

• Time dilation: A moving clock ticks slower than a stationary one relative to an observer.
• Length contraction: Objects in motion appear shorter in the direction of motion to an observer.
• Mass increase: As an object moves closer to the speed of light, its mass appears to increase from an external viewpoint.

In cases like our light frequency shift, relativistic effects significantly alter observations. The Doppler effect in relativity doesn't just move the apparent frequency but changes how we fundamentally perceive that motion. For instance, if a car was indeed moving fast enough to change perceived light colors, it would illustrate these relativistic phenomena. By using the principles of relativity, we can calculate these moving frame effects accurately, showing the beauty and depth of modern physics.