Question

A source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

Short answer

The source is moving toward you at approximately 21.96% of the speed of light.

Step-by-step solution

Understand the Doppler Effect Formula

The frequency observed, \( f' \), is related to the rest frequency, \( f \), by the relativistic Doppler effect. The formula is \( f' = f \sqrt{\frac{1 + \beta}{1 - \beta}} \), where \( \beta = \frac{v}{c} \) and \( v \) is the velocity of the source, while \( c \) is the speed of light.

Rearrange the Doppler Effect Formula

Since \( f' = 1.25f \), substitute in the formula. We get \( 1.25 = \sqrt{\frac{1 + \beta}{1 - \beta}} \). To solve for \( \beta \), square both sides: \( 1.5625 = \frac{1 + \beta}{1 - \beta} \).

Solve for Beta (\(\beta\))

Cross-multiply to eliminate the fraction: \( 1.5625(1 - \beta) = 1 + \beta \). Expanding gives \( 1.5625 - 1.5625\beta = 1 + \beta \). Combine like terms: \( 0.5625 = 2.5625\beta \). Hence, \( \beta = \frac{0.5625}{2.5625} \approx 0.2196 \).

Interpret the Velocity Result

Since \( \beta < 1 \) and positive, this implies \( v = 0.2196c \), or 21.96% of the speed of light. Since the observed frequency is higher, the source is moving toward you.

Key concepts

Electromagnetic Radiation

Electromagnetic radiation refers to waves of energy associated with electric and magnetic fields. These waves travel through space at the speed of light, which is approximately \(3 \times 10^8\) meters per second.
Electromagnetic waves include a wide range of types, such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each type has different properties, but all share the same fundamental nature of being electromagnetic radiation.
Understanding electromagnetic radiation is essential because it forms the basis of many technologies and scientific explorations. For example, light from stars, radio broadcasts, and microwaves for cooking are all forms of electromagnetic radiation. This concept plays a crucial role in fields ranging from astronomy to communications.

Velocity Calculation

Calculating the velocity of a moving source in the context of the Doppler effect involves determining the speed at which the source is approaching or receding from the observer. This is done using the relativistic Doppler effect formula.
The formula relates the observed frequency \( f' \) to the rest frequency \( f \), using the equation \( f' = f \sqrt{\frac{1 + \beta}{1 - \beta}} \), where \( \beta = \frac{v}{c} \). Here, \( v \) is the velocity of the source and \( c \) is the speed of light.
To find the velocity \( v \), we first solve for \( \beta \). This process involves:

• Substituting the known values into the formula.
• Rearranging and solving the equation for \( \beta \).
• Multiplying \( \beta \) by the speed of light \( c \) to find \( v \).

In our solution, we found \( \beta \) to be approximately 0.2196, indicating that the source is moving at 21.96% the speed of light toward the observer.

Frequency Shift

A frequency shift occurs when the frequency of a wave changes due to the motion of the source relative to the observer. In the context of the Doppler effect, this shift can either increase or decrease, depending on whether the source is moving toward or away from the observer.
A higher observed frequency compared to the rest frequency indicates that the source is moving toward the observer. Conversely, a lower observed frequency suggests that the source is moving away.
This concept is crucial for understanding the implications of motion on wave properties. For instance, astronomers use frequency shifts to determine the motion of stars and galaxies. A shift to higher frequencies (blue shift) signifies an approaching astronomical object, while a shift to lower frequencies (red shift) indicates a receding object. The calculated frequency shift in our exercise shows the source is moving closer, as the observed frequency is 1.25 times the rest frequency.