Doppler Effect Worksheet Answers: Clear Insights & Solutions
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The Doppler Effect is a fascinating phenomenon in physics that describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. This effect is not only relevant in sound waves but also in light waves, offering numerous applications in various fields, such as astronomy and radar technology. In this article, we will dive deep into the concept of the Doppler Effect and provide clear insights, as well as solutions, to common worksheet questions related to this intriguing topic.
Understanding the Doppler Effect
What is the Doppler Effect?
The Doppler Effect is observed when there is a relative motion between a wave source and an observer. When the source of the waves moves towards the observer, the waves get compressed, leading to a higher frequency. Conversely, when the source moves away, the waves stretch, resulting in a lower frequency. This can be easily illustrated with sound waves:
- Approaching Source: Higher pitch sound. ๐
- Receding Source: Lower pitch sound. ๐
Real-world Examples
- Emergency Vehicles: When an ambulance approaches, its siren sounds higher in pitch. As it moves away, the pitch drops noticeably.
- Astronomy: The Doppler Effect helps astronomers determine whether stars or galaxies are moving towards or away from Earth, affecting the light we receive.
Mathematical Formulation
The Doppler Effect can be mathematically expressed in both sound and light waves.
For Sound Waves
When the source of sound is moving towards a stationary observer, the observed frequency (( f' )) can be calculated using the formula:
[ f' = f \left( \frac{v + v_o}{v - v_s} \right) ]
Where:
- ( f' ) = observed frequency
- ( f ) = emitted frequency
- ( v ) = speed of sound in the medium
- ( v_o ) = speed of the observer (positive if moving towards the source)
- ( v_s ) = speed of the source (positive if moving away from the observer)
For Light Waves
The Doppler Effect for light can be expressed as:
[ f' = f \sqrt{\frac{1 + \beta}{1 - \beta}} ]
Where ( \beta ) is defined as the ratio of the relative velocity of the source and observer to the speed of light.
Common Worksheet Questions and Solutions
Now let's address some common questions and their solutions that may be encountered in a Doppler Effect worksheet.
Question 1: An ambulance with a siren frequency of 500 Hz is moving towards a stationary observer at a speed of 30 m/s. Calculate the observed frequency. Assume the speed of sound in air is 343 m/s.
Solution: Using the formula for sound waves:
[ f' = f \left( \frac{v + v_o}{v - v_s} \right) ]
Substituting the values:
- ( f = 500 , \text{Hz} )
- ( v = 343 , \text{m/s} )
- ( v_o = 0 , \text{m/s} ) (stationary observer)
- ( v_s = 30 , \text{m/s} )
[ f' = 500 \left( \frac{343 + 0}{343 - 30} \right) \approx 500 \left( \frac{343}{313} \right) \approx 548 , \text{Hz} ]
Question 2: A star is moving away from Earth at a velocity of 2000 km/s. If the emitted light has a wavelength of 500 nm, what is the observed wavelength?
Solution: Using the formula for light waves, first convert the speed from km/s to m/s:
[ \beta = \frac{2000 \times 10^3}{3 \times 10^8} = 0.00667 ]
Now, substituting into the wavelength formula (( \lambda' = \lambda \sqrt{\frac{1 + \beta}{1 - \beta}} )):
[ \lambda' = 500 \left( \sqrt{\frac{1 + 0.00667}{1 - 0.00667}} \right) \approx 500 \left( \sqrt{\frac{1.00667}{0.99333}} \right) \approx 500 \times 1.0067 \approx 503.35 , \text{nm} ]
Summary Table of Key Concepts
<table> <tr> <th>Concept</th> <th>Sound Waves</th> <th>Light Waves</th> </tr> <tr> <td>Frequency Change</td> <td>Higher when approaching, lower when receding</td> <td>Redshift (moving away), blueshift (moving towards)</td> </tr> <tr> <td>Mathematical Formula</td> <td>f' = f (v + v_o)/(v - v_s)</td> <td>f' = f โ((1 + ฮฒ)/(1 - ฮฒ))</td> </tr> <tr> <td>Applications</td> <td>Radar, ambulance sirens</td> <td>Astronomy, spectroscopy</td> </tr> </table>
Important Notes
"The Doppler Effect has significant implications in various fields. Understanding the mathematical applications can help in a variety of practical situations, from emergency responses to exploring the cosmos."
In conclusion, the Doppler Effect is a vital concept that connects our everyday experiences with advanced scientific applications. By grasping this phenomenon, we open the door to a deeper understanding of waves and their behaviors in different contexts. The worksheets provide excellent opportunities to engage with these concepts through practical problems that enhance learning. Whether you're a student or a curious learner, mastering the Doppler Effect can enrich your appreciation of the world around you.