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Magazine Chapter 12: Problem 59
How fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we receive from it is \(10.0 \%\) higher than the frequency of the light it is emitting? Would it be moving away from us or toward us? (Assume it is moving either directly away from us or directly toward us.)
Short Answer
Expert verified
The star moves toward us at approximately 9.5% of the speed of light.
Step by step solution
01
Understand the Doppler Effect
The Doppler effect refers to changes in the frequency of waves, such as sound or light, due to the relative motion of the source and the observer. For light, if the source moves towards the observer, the frequency increases (blueshift); if it moves away, the frequency decreases (redshift).
02
Apply the Frequency Formula
When the frequency received \( f' \) is increased by 10%, it means \( f' = 1.10 \, f \), where \( f \) is the emitted frequency. The formula for the Doppler effect regarding light is: \[ f' = f \left( \frac{1 + \frac{v}{c}}{1 - \frac{v}{c}} \right)^{1/2} \]where \( v \) is the velocity of the star and \( c \) is the speed of light.
03
Solve the Doppler Effect Equation
To find the speed as a percentage of the light speed, solve:\[ 1.10 \cdot f = f \left( \frac{1 + \frac{v}{c}}{1 - \frac{v}{c}} \right)^{1/2} \]Divide both sides by \( f \):\[ 1.10 = \left( \frac{1 + \frac{v}{c}}{1 - \frac{v}{c}} \right)^{1/2} \]Square both sides:\[ 1.21 = \frac{1 + \frac{v}{c}}{1 - \frac{v}{c}} \]
04
Simplify and Solve for \( v/c \)
Multiply both sides by \( 1 - \frac{v}{c} \):\[ 1.21 \left( 1 - \frac{v}{c} \right) = 1 + \frac{v}{c} \]Expand and simplify:\[ 1.21 - 1.21 \frac{v}{c} = 1 + \frac{v}{c} \]Rearrange the terms:\[ 1.21 - 1 = 1.21 \frac{v}{c} + \frac{v}{c} \]\[ 0.21 = 2.21 \frac{v}{c} \]
05
Calculate \( v/c \) Value
Solve for \( \frac{v}{c} \):\[ \frac{v}{c} = \frac{0.21}{2.21} \]
06
Final Calculation and Interpretation
Calculate the numerical value:\[ \frac{v}{c} \approx 0.095 \]This means the star is moving at approximately 9.5% of the speed of light. Since the frequency is higher than emitted, the star is moving toward us.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Frequency Change
The concept of frequency change is essential in understanding how motion affects light or sound waves. When an object emitting waves moves towards an observer, the waves become compressed, leading to an increased frequency. This is known as a "frequency increase," or blueshift. Conversely, if the object moves away, the waves are stretched, causing a "frequency decrease," or redshift.
This change helps scientists understand the relative velocity and direction of celestial objects.
- When the frequency of light increases, we say it's "blueshifted."
- A decrease in frequency results in a "redshift."
This change helps scientists understand the relative velocity and direction of celestial objects.
Light Waves
Light waves are electromagnetic waves that can travel through a vacuum, unlike sound waves which need a medium. They have properties like wavelength, frequency, and speed, which help explain how they interact with different environments. As objects move relative to an observer, the observed frequency of these light waves can change due to their intrinsic properties.
- Light waves move at a constant speed in a vacuum鈥� the speed of light.
- Wavelengths of light waves can vary from longer wavelengths (red light) to shorter ones (blue light).
Speed of Light
The speed of light, denoted as \( c \), is a fundamental constant of nature. It represents the maximum speed at which all energy, matter, and information in the universe can travel. In a vacuum, light travels at approximately \( 299,792,458 \) meters per second (or about 300,000 kilometers per second).
- Light speed is constant and forms a basis for many scientific calculations.
- It's denoted by \( c \) and is crucial in relativity equations.
Blueshift
Blueshift occurs when the frequency of the light from an object increases as it moves towards the observer. This higher frequency results in light appearing bluer on the color spectrum. In the exercise, we determined the star emits light with a 10% frequency increase.
- Blueshift indicates motion towards the observer.
- It is commonly seen in astronomy when stars or galaxies move closer to us.
Relative Motion
Relative motion plays a critical role in the Doppler Effect observed for light waves. It describes how the velocity of an object is considered concerning the observer's position and motion. In our scenario, understanding whether a star moves towards or away from Earth affects how we perceive its emitted light.
- Relative motion modifies the observed frequency of light.
- It governs the shift in wavelengths, causing either blueshift or redshift.
