Problem 99 - Fundamentals of Physics

Question

A spaceship approaches Earth at a speed of \(0.42 c .\) A light on the front of the ship appears red (wavelength \(650 \mathrm{nm}\) ) to passengers on the ship. What (a) wavelength and (b) color (blue, green, or yellow) would it appear to an observer on Earth?

Step-by-Step Solution

Answer

The light would appear with a wavelength of 415.5 nm and appear blue to an Earth observer.

Step 1: Understanding the Problem

We have a spaceship moving towards Earth with a speed of \(0.42c\), where \(c\) is the speed of light. Passengers on the ship see the light with a wavelength of \(650\ \text{nm}\). We need to find the wavelength of light as seen by an observer on Earth using the Doppler effect for light.

Step 2: Identifying the Doppler Effect Formula

For light moving towards an observer, the relativistic Doppler shift formula is \( \lambda = \lambda_0 \sqrt{\frac{1-\beta}{1+\beta}} \), where \(\lambda_0\) is the original wavelength, \(\lambda\) is the observed wavelength, and \(\beta = \frac{v}{c}\) is the ratio of the object's speed to the speed of light.

Step 3: Calculating Beta

We are given that the spaceship is moving at \(0.42c\). Thus, \(\beta = 0.42\).

Step 4: Calculating the Observed Wavelength

Substitute the given values into the formula: \[ \lambda = 650\ \text{nm} \times \sqrt{\frac{1-0.42}{1+0.42}} \] Calculating \(\sqrt{\frac{1-0.42}{1+0.42}}\), we find \(\sqrt{\frac{0.58}{1.42}} = \sqrt{0.4085} \approx 0.6393\). Thus, \(\lambda \approx 650 \times 0.6393 = 415.545\ \text{nm}\).

Step 5: Determining the Color of the Observed Wavelength

The wavelength of \(415.545\ \text{nm}\) falls within the range of blue light in the visible spectrum. Therefore, the light appears blue to the observer on Earth.

Key Concepts

Relativistic Effects

When objects move at speeds close to that of light, fascinating things happen due to relativistic effects. These effects arise from Einstein's theory of relativity and radically alter how we perceive speed, time, and light.

In our exercise, the spaceship is moving at a substantial fraction of the speed of light, specifically at 0.42 times the speed of light. This introduces the relativistic Doppler effect, which affects the observed wavelength of light. When considering relativistic effects:

• Speeds are compared to the speed of light ( c ).
• The relativistic Doppler shift modifies how light is perceived by observers moving relative to the source.
• Equations must account for these effects to accurately describe physical phenomena.

By using the formula associated with these effects, we can determine changes in the perception of light from different frames of reference, like that of an Earth observer compared to a spaceship passenger.

Wavelength Shift

The phenomenon of wavelength shift is critical to understanding how light changes its properties when observed from different perspectives. Wavelength is inversely proportional to frequency; when one increases, the other decreases. As a spaceship moves closer to Earth at a high speed, the light waves it emits are compressed.

This compression results in a decrease in the observed wavelength, known in this context as a blueshift. The Doppler effect can therefore make the same light appear to be of different wavelengths due to motion.

In step-by-step calculation, we saw this in motion:

• Initial wavelength: 650 nm (appearing red in the spaceship).
• Observed wavelength: Calculated as approximately 415.545 nm (appearing blue to observers on Earth).

Understanding how both speed and direction influence wavelength shifts is crucial in fields like astronomy and satellite communications.

Visible Spectrum

The visible spectrum refers to the range of light wavelengths that the human eye can perceive. This spectrum typically ranges from about 380 nm (violet) to 750 nm (red). When light enters this range from different parts of the electromagnetic spectrum due to motion like the ship's, it can change colors.

In the original problem, a light appearing red from the perspective of the ship's passengers shifts to blue for an observer on Earth after undergoing a relativistic Doppler shift. This happens because:

• The initial wavelength shrinks into the shorter wavelengths of the blue range.
• The calculated new wavelength makes it detectable as blue, around 415 nm, within our visible spectrum.

Such changes are vital for astronomers, especially when analyzing the movement of stars and galaxies, helping them determine velocities and directions relative to Earth.