Changes due to the movement of emitter and receiver (the Doppler effect)

An ambulance with its siren on approaches a pedestrian, passes them and moves away.

1. a) Describe how the pitch perceived by the pedestrian changes as the ambulance approaches and as it moves away (higher or lower pitch).
2. b) Explain the change using the compression and spreading out of the wavefronts, using a diagram in your notebook to help you.
3. c) Does the frequency actually emitted by the siren change? Distinguish between emitted frequency and perceived frequency.

When a source approaches a stationary observer, the observer perceives the frequency $f' = f \cdot \frac{v}{v - v_f}$ (formula given; speed of sound in air: $v = 340\ \text{m/s}$).

An ambulance emits sound at $500\ \text{Hz}$ and approaches a pedestrian at $20\ \text{m/s}$. Calculate the frequency perceived by the pedestrian.

When the observer moves towards a stationary source, the formula is $f' = f \cdot \frac{v + v_0}{v}$ (formula given; $v = 340\ \text{m/s}$).

A cyclist rides at $10\ \text{m/s}$ towards a loudspeaker emitting a $600\ \text{Hz}$ tone. The figure shows the situation.

Calculate the frequency perceived by the cyclist.

If both the source and the observer move towards each other at the same time, the perceived frequency is $f' = f \cdot \frac{v + v_0}{v - v_f}$ (formula given; $v = 340\ \text{m/s}$).

A train sounds a $400\ \text{Hz}$ whistle as it approaches at $30\ \text{m/s}$, while a cyclist pedals at $10\ \text{m/s}$ towards it. The figure shows the situation.

Calculate the frequency perceived by the cyclist.

An ambulance emits sound at $600\ \text{Hz}$ and travels at $72\ \text{km/h}$ ($v = 340\ \text{m/s}$).

1. a) Express the speed in $\text{m/s}$.
2. b) Calculate the frequencies perceived by a stationary pedestrian as it approaches and as it moves away.
3. c) Determine the drop in frequency perceived just as it passes, and use it to explain the characteristic change in pitch of sirens when they overtake us.
4. d) Draw in your notebook the wavefronts of a moving source and indicate where they bunch up and where they spread apart.

You can do it on paper and send a photo with your answer.

When the source moves away from a stationary observer, the formula is $f' = f \cdot \frac{v}{v + v_f}$ (formula given; $v = 340\ \text{m/s}$).

1. a) Using the same ambulance as before ($f = 500\ \text{Hz}$, $v_f = 20\ \text{m/s}$), calculate the frequency perceived as it moves away.
2. b) Verify that it is lower than the emitted frequency and explain it qualitatively using the spacing of the wavefronts.

An ambulance emits sound at $700\ \text{Hz}$ and travels at $90\ \text{km/h}$ (speed of sound: $v = 340\ \text{m/s}$).

1. a) Express the speed of the ambulance in $\text{m/s}$.
2. b) Calculate the frequency perceived by a stationary pedestrian as it approaches, using $f' = f \cdot \frac{v}{v - v_f}$.
3. c) Calculate it as it moves away, using $f' = f \cdot \frac{v}{v + v_f}$.

The radial velocity method detects exoplanets from the wobble they induce in their star: spectral lines oscillate periodically with $\frac{\Delta \lambda}{\lambda} = \frac{v}{c}$ (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$). In the $\lambda = 656,3\ \text{nm}$ line of a star, a maximum oscillation of $\Delta \lambda = 1,3 \cdot 10^{-4}\ \text{nm}$ is measured. The diagram shows the wobble of the star.

1. a) Express both lengths in metres and calculate the maximum speed of the star.
2. b) Explain why this periodic oscillation reveals an invisible planet that forces the star to orbit the common centre of mass of the system.
3. c) Reason what additional information about the planet the period of oscillation would provide.

In a Doppler ultrasound scan of a blood vessel, the frequency shift of the echo is $\Delta f \approx \frac{2 \cdot f \cdot v_{\text{sangre}}}{v}$ (formula given; speed of sound in tissue: $v = 1540\ \text{m/s}$). The probe emits at $5\ \text{MHz}$ and the blood flows at $0,3\ \text{m/s}$.

1. a) Calculate $\Delta f$.
2. b) Compare the result with the range of human hearing ($20\ \text{Hz}$ to $20\ \text{kHz}$) and reason why the equipment converts this shift into an audible tone for the doctor.

The H$\alpha$ line of hydrogen ($\lambda_0 = 656,3\ \text{nm}$ in the laboratory) is observed in the spectrum of a galaxy at $\lambda' = 678,9\ \text{nm}$ (data: $z = \frac{\Delta \lambda}{\lambda_0}$; $v \approx z \cdot c$; Hubble's law $v = H_0 \cdot d$, with $H_0 = 70\ \text{km/s}$ per $\text{Mpc}$; $c = 3 \cdot 10^{5}\ \text{km/s}$).

1. a) Calculate the redshift $z$ of the galaxy.
2. b) Calculate its speed in $\text{km/s}$.
3. c) Calculate its distance in $\text{Mpc}$.
4. d) Explain what cosmological consequence is deduced from the fact that distant galaxies systematically show $z > 0$.

A weather radar emits at $5,6 \cdot 10^{9}\ \text{Hz}$ and measures the echo from raindrops using $\Delta f \approx \frac{2 \cdot f \cdot v}{c}$ (formula given; $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate $\Delta f$ for raindrops moving away from the radar at $15\ \text{m/s}$.
2. b) State whether the echo returns with a higher or lower frequency than the emitted frequency and justify your answer.
3. c) Explain how, by sweeping the beam in different directions, the radar reconstructs a wind map inside a storm.