Every formula, sign convention, trick, and trap for IB Physics HL Topic C.5 Doppler Effect — moving source and observer formulas, light Doppler shift, redshift, and applications from radar to Hubble.
The Doppler Effect closes the C-block of IB Physics HL by linking wave physics to astrophysics, medical imaging and traffic enforcement. SL students need the qualitative idea (approaching $\Rightarrow f' > f$, receding $\Rightarrow f' < f$) and the simple light-Doppler approximation $\Delta f / f = v/c$. HL adds the quantitative sound formulas for moving source and moving observer, where the asymmetry between the two cases is one of the most-examined details in Topic C.5.
This cheatsheet condenses every Topic C.5 formula and exam trap into one revision sheet — wavefront diagrams, the four sound-Doppler cases, redshift vs blueshift, and the radar / ultrasound / Hubble applications. The most common HL pitfalls — mixing the source and observer formulas, getting the sign wrong, missing the factor of 2 in the radar formula, applying the simple $v/c$ relation to sound — are flagged in red.
The Doppler effect: the observed frequency of a wave changes when the source and observer move relative to each other.
Approaching $\Rightarrow f' > f$ (higher frequency, shorter wavelength).
Receding $\Rightarrow f' < f$ (lower frequency, longer wavelength).
The speed of the wave in the medium is unchanged by source or observer motion.
Wavefront summary
In front of source
Behind source
Wavelength
Compressed ($\lambda' < \lambda$)
Stretched ($\lambda' > \lambda$)
Frequency
Higher ($f' > f$)
Lower ($f' < f$)
Speed of sound
Same as stationary
Same as stationary
NoteFor a moving source, the medium has compressed wavefronts in front and stretched wavefronts behind. For a moving observer, the wavefront spacing is unchanged — only the rate at which the observer intercepts wavefronts differs.
The Doppler effect. A moving source crowds its wavefronts ahead of it (observer A hears a higher frequency) and stretches them behind (observer B hears lower) — the wave speed itself is unchanged.
The wave speed in the medium is UNCHANGED by source/observer motion — only $\lambda$ (moving source) or the interception rate (moving observer) changes.
A moving source bunches wavefronts in front and stretches them behind.
Observer towards:$f' = f \cdot \dfrac{v + u_o}{v}$
Observer away:$f' = f \cdot \dfrac{v - u_o}{v}$
where $v$ is the speed of sound in the medium, $u_s$ is the source speed, and $u_o$ is the observer speed (both relative to the medium).
TrickTowards = $f$ goes UP. Moving source towards: subtract $u_s$ in the denominator (smaller denominator $\Rightarrow$ larger $f'$). Moving observer towards: add $u_o$ in the numerator. Always sanity-check at the end: approach $\Rightarrow$ answer $> f$; recede $\Rightarrow$ answer $< f$.
TrapIB problems will never have both source and observer moving simultaneously — it is always one or the other. The two formulas give different numerical answers for the same speed $u$, so do not mix them up.
NoteTo find a speed from an observed frequency, rearrange the formula. Example, moving observer away: $u_o = v\left(1 - \dfrac{f'}{f}\right)$.
Source vs observer. A moving source uses $f'=f\,v/(v-u_s)$; a moving observer uses $f'=f(v+u_o)/v$ — for approach, both raise the frequency.
From the Photon question bank
What examiners actually test on this concept:
Source: v/(v∓u_s)Observer: (v±u_o)/vOne or the otherRearrange for speed
Towards ⇒ $f$ goes UP. Moving source: subtract $u_s$ in the denominator; moving observer: add $u_o$ in the numerator. very frequent (HL)
IB never moves source AND observer at once — always one or the other; don't mix the two formulas.
Always sanity-check (approach ⇒ $f'>f$) and rearrange to find a speed from $f$ and $f'$.
For light or any EM wave, when the source–observer relative speed $v$ is much less than $c$ (the non-relativistic limit), the source-vs-observer asymmetry vanishes and a single formula applies:
Frequency shift:$\dfrac{\Delta f}{f} \approx \dfrac{v}{c}$
TrickSpectral line shift: compare the observed line position with the known rest wavelength. If the line is shifted to the right (longer $\lambda$) $\Rightarrow$ redshift $\Rightarrow$ receding. Shifted to the left (shorter $\lambda$) $\Rightarrow$ blueshift $\Rightarrow$ approaching.
TrapThe simple $\Delta f/f \approx v/c$ formula applies only to EM waves in the non-relativistic limit. For sound, the moving-source vs moving-observer asymmetry matters and you must use the $f' = f \cdot v / (v \mp u_s)$ family of formulas.
Redshift and blueshift. Against the rest wavelength, an approaching source shifts the line toward the blue (shorter $\lambda$) and a receding one toward the red (longer $\lambda$).
From the Photon question bank
What examiners actually test on this concept:
Δf/f ≈ v/cRedshift / blueshiftv = c·Δλ/λEM only (not sound)
For EM in the non-relativistic limit one formula: $\Delta f/f \approx \Delta\lambda/\lambda \approx v/c$ (no source/observer asymmetry). frequent
The $v/c$ formula is EM only — sound keeps the moving-source vs moving-observer asymmetry.
§4 — Applications C.5 SL + HL
Radar speed gun: microwaves reflect off a moving car; the frequency shift $\Delta f = 2 v f / c$ gives the car's speed. The factor of 2 arises because the car acts as both moving observer (on receiving) and moving source (on re-emitting).
Medical ultrasound: sound reflects off moving blood cells; the frequency shift gives the blood-flow speed. Used in foetal monitoring and cardiology.
Stellar / galactic motion: spectral line shift gives a star's recession or approach velocity along the line of sight.
Hubble expansion: all distant galaxies are redshifted, demonstrating that the universe is expanding ($v \approx H_0 d$).
NoteFor medical ultrasound, the probe acts as both source and detector. The frequency shift is $\Delta f = 2 v f \cos\theta / c_\text{tissue}$, where $\theta$ is the angle between the beam and the blood-flow direction. Maximum shift occurs at $\theta = 0$ (beam parallel to flow).
Radar speed gun. Microwaves reflect off a moving car and come back shifted twice, so $\Delta f = 2vf/c$ gives the speed.
From the Photon question bank
What examiners actually test on this concept:
Radar Δf = 2vf/cThe factor of 2Ultrasound cos θHubble expansion
Radar speed gun: $\Delta f=2vf/c$ — the factor of 2 is because the car is both a moving observer (receiving) and a moving source (re-emitting). frequent
Medical ultrasound: $\Delta f = 2vf\cos\theta/c_{\text{tissue}}$; maximum shift when the beam is parallel to the flow ($\theta=0$).
Galaxy redshifts give $v \approx H_0 d$ — evidence the universe is expanding.
§5 — Exam Attack Plan All sections
When you see this in the question — reach for that:
Question trigger
Reach for
Moving source, find $f'$
$f' = f \cdot v / (v \mp u_s)$ (subtract for approach).
Moving observer, find $f'$
$f' = f \cdot (v \pm u_o) / v$ (add for approach).
Wavelength change for moving source
$\lambda' = (v \mp u_s) / f$.
Find observer or source speed from $f, f'$
Rearrange the appropriate formula.
Light from a distant galaxy
$\Delta\lambda/\lambda \approx v/c \Rightarrow v = c\,\Delta\lambda/\lambda$.
Redshift or blueshift?
Compare $\lambda_\text{obs}$ and $\lambda_\text{rest}$.
Radar speed gun
$\Delta f = 2 v f / c$ (factor of 2!).
Medical ultrasound
$\Delta f = 2 v f \cos\theta / c_\text{tissue}$.
"Why are spectral lines shifted?"
Source–observer relative motion + Doppler effect.
Sanity check
Approaching $\Rightarrow f' > f$; receding $\Rightarrow f' < f$.
Worked Example — IB-Style HL Doppler Effect Problem
Question (HL Paper 2 style — 7 marks)
A police car siren emits a sound of frequency 750 Hz. The car moves directly towards a stationary observer at 25 m s${}^{-1}$. The speed of sound in air is 340 m s${}^{-1}$. (a) Calculate the frequency heard by the observer. (b) After passing the observer, the car continues at the same speed. Calculate the new observed frequency. (c) State, with reason, what would change if instead the car were stationary and the observer moved towards it at 25 m s${}^{-1}$.
Solution
Source moving towards observer — use $f' = f \cdot v / (v - u_s)$ with $v = 340$, $u_s = 25$. (M1)
After passing, source moving away — use $f' = f \cdot v / (v + u_s)$. $f' = 750 \cdot 340 / 365 = 698.6\ \mathrm{Hz} \approx 699\ \mathrm{Hz}$. (M1)(A1)
For a moving observer towards a stationary source: $f' = f \cdot (v + u_o) / v = 750 \cdot 365 / 340 = 805.1\ \mathrm{Hz} \approx 805\ \mathrm{Hz}$. (M1)(A1)
The observed frequency is different (805 Hz vs 810 Hz) even though the relative speed is the same. This is because the source's motion compresses wavefronts in the medium, while a moving observer simply intercepts unchanged wavefronts at a different rate. (R1)
Examiner's note: The most common error is using the wrong formula — putting $u_s$ in the numerator or $u_o$ in the denominator. Always sketch the situation, identify which is moving (source vs observer), and apply the correct formula. The asymmetry (different answers for the same speed) is the conceptual point Topic C.5 is testing.
Common Student Questions
Is the moving-source formula different from the moving-observer formula?
Yes. Moving source: $f' = f \cdot v / (v \mp u_s)$ — the change is in the denominator. Moving observer: $f' = f \cdot (v \pm u_o) / v$ — the change is in the numerator. The two formulas give different numerical answers for the same speed $u$, so you must identify which case applies. IB exam questions never have both source and observer moving simultaneously — it is always one or the other.
How do I know which sign to use in the Doppler formula?
Approaching always increases the observed frequency, receding always decreases it. Moving source towards observer: subtract $u_s$ in the denominator (smaller denominator means $f' > f$). Moving source away: add $u_s$. Moving observer towards source: add $u_o$ in the numerator. Moving observer away: subtract $u_o$. Always sanity-check at the end: if approaching, your answer must be greater than $f$; if receding, less than $f$.
What is the difference between redshift and blueshift?
Redshift: the observed wavelength is longer than the rest wavelength ($\lambda_\text{obs} > \lambda_\text{rest}$), so the spectral lines shift towards the red end of the spectrum. This happens for receding sources — distant galaxies are all redshifted (Hubble's law). Blueshift: observed wavelength is shorter than the rest wavelength, spectral lines shift towards blue. This happens for approaching sources — the Andromeda galaxy is blueshifted because it is moving towards the Milky Way.
Why is there a factor of 2 in the radar speed-gun formula?
The radar undergoes two Doppler shifts: once when the moving car receives the wave (acting as a moving observer), and once when it re-emits the wave (now acting as a moving source). Each shift contributes $\Delta f = v f / c$, so the total shift is $\Delta f = 2 v f / c$. This is why the IB-given formula for radar speed measurement carries the factor of 2 — and why missing it halves the calculated speed.
When can I use the simple $\Delta f / f = v/c$ formula?
It is valid for electromagnetic waves (light, radio, microwaves) when the source–observer relative speed $v$ is much less than $c$ (the non-relativistic limit). The same approximate equality holds for fractional wavelength shift: $\Delta\lambda/\lambda \approx v/c$. Do not use it for sound — sound has a medium (air) and the moving-source vs moving-observer asymmetry matters; use the $f' = f \cdot v / (v \mp u_s)$ formulas instead.
What's NOT in this cheatsheet
This page gives you the formulas and the traps. The full Photon Academy Doppler Effect library (only available to enrolled students or via the resource library subscription) adds:
Topic C.5 Notes PDF — every concept worked through in full, with derivations of the moving-source and moving-observer formulas.
Tutorial booklet — 25+ IB-style questions sequenced from foundation to AHL difficulty, including two-step radar/echo problems.
Tutorial Solutions — full mark-scheme-style worked solutions with M1/A1/R1 annotations.
Practice Solutions — extra past-paper-style problems with detailed walk-throughs.
Cheatsheet PDF — print-ready, brand-formatted, the same one our students take into mock exams.
