Infer recession speed from a supplied redshift relation
Explain how a wavelength change can be used to find a distant galaxy’s recession speed.
Extended only (Supplement); Physics 0625, 2026–2028. Use supplied distances, periods and stated orbital models. Drawings are not to scale. Written practice is self-checked.
Before you begin
- Compare corresponding spectral wavelengths.
- Use scientific notation and ratios.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the unit of a ratio of two common-unit wavelengths?
What you will learn
- Explain how a wavelength change can be used to find a distant galaxy’s recession speed.
- Apply a supplied small-redshift approximation with units and an explicit validity boundary.
Move from observation to a speed model
The change between observed and reference starlight wavelengths supplies information used to determine recession speed. A larger fractional shift corresponds to a larger inferred recession speed within a stated model. A wavelength change in nanometres by itself is not a speed in metres per second.
For practice only, the supplied small-redshift approximation is v/c ≈ (λobserved − λreference)/λreference, with c = 3.0 × 10⁸ m/s. The syllabus requires knowing the wavelength-speed link; it does not require recalling this extra approximation as a universal formula.
Keep the approximation honest
Compute the wavelength difference and divide by the reference wavelength using common wavelength units. That fraction is dimensionless; multiplying it by c gives speed. Replacing the reference denominator by the observed wavelength changes the calculation.
The relation is approximate for small redshift. Large cosmological redshifts need an appropriate cosmological model, not an unqualified extension of this calculation. Apparent source brightness does not substitute for the matching wavelength comparison.
Pause and explain
What is needed to turn a wavelength shift into speed?
Worked example
Using the supplied small-shift relation, λreference = 500 nm and λobserved = 505 nm. Find v.
Show the worked solution
- Δλ = 5 nm.
- Δλ/λreference = 5/500 = 0.01.
- v ≈ 0.01 × 3.0 × 10⁸ m/s.
Answer Approximately 3.0 × 10⁶ m/s, or 3000 km/s.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Use the supplied approximation for reference 600 nm and observed 606 nm. Find the recession speed.
Give me a hint
Divide the matching increase by the reference before multiplying by c.
Compare my reasoning
- Δλ = 6 nm.
- Fraction = 6/600 = 0.01.
- v ≈ 0.01 × 3.0 × 10⁸.
3.0 × 10⁶ m/s.
Look for these in your work
- Use the reference denominator.
- State the approximation.
For reference 400 nm and observed 402 nm, use the same approximation.
Give me a hint
Keep both wavelengths in the same unit.
Compare my reasoning
- Δλ = 2 nm.
- Fraction = 2/400 = 0.005.
- v ≈ 0.005 × 3.0 × 10⁸.
1.5 × 10⁶ m/s, or 1500 km/s.
Look for these in your work
- Compute the fractional shift.
- Convert km/s consistently.
A learner applies the small-shift formula to a huge wavelength ratio and reports an exact cosmic speed. Evaluate it.
Give me a hint
Check the validity of the supplied model before trusting the number.
Compare my reasoning
- The formula is a small-redshift approximation.
- A large shift is outside its stated use.
- A suitable cosmological relation is needed.
The exact-speed claim is unsupported; the approximation’s stated validity has been exceeded.
Look for these in your work
- Identify the model limit.
- Distinguish arithmetic from validity.
Common mistakes
- A wavelength change is not directly a speed.
- Use reference wavelength in the supplied ratio.
- Small-redshift approximation is not universally exact.
What can you explain now?
What is needed to turn a wavelength shift into speed?
Compare with the explanation
A stated wavelength-speed relation
Spectral data supply the shift, and the appropriate model relates it to recession speed.
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Make it stick
Apply a supplied small-redshift approximation with units and an explicit validity boundary.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.