Calculate Hubble constant from recession speed and distance
Define H₀ as recession speed divided by galaxy distance and use consistent SI units.
Section P5.4 · The Universe
Cross-referenced to the 2028–2029 0653 section using original teaching from the related single-science courses. A 0653-specific objective-by-objective and Core/Supplement review is in progress. Written practice is self-checked and does not demonstrate supervised laboratory competence or award an official Cambridge grade.
Before you begin
- Convert kilometres to metres.
- Divide quantities written in scientific notation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which speed value is 6600 km/s in m/s?
What you will learn
- Define H₀ as recession speed divided by galaxy distance and use consistent SI units.
- Rearrange H₀ = v/d to find the ratio, speed or distance.
Define the measured ratio
The Hubble constant H₀ is the ratio of a galaxy’s recession speed v to its distance d: H₀ = v/d. In the simple far-galaxy relation, v = H₀d and d = v/H₀. Use the distance and speed that belong to the same galaxy and the stated model.
With v in m/s and d in metres, metres cancel and H₀ has unit s⁻¹, meaning per second. It is neither a speed in m/s nor a time in seconds. Mixing a speed number in km/s with a metre distance introduces a factor-of-1000 error.
Name the interpretation boundary
The relation describes a large-scale trend for far galaxies; local motions can contribute additional speed. A supplied pair yields a model ratio, not proof that one galaxy alone establishes the whole cosmic expansion history.
A larger distance at the same H₀ corresponds to a larger recession speed. This proportionality is not a claim that brightness directly equals speed or that physical star sizes grow by H₀ every second. Keep the measured quantities and their units distinct.
Sources for this lesson
Pause and explain
What is the SI unit of v/d?
Worked example
Find H₀ for supplied v = 6600 km/s and d = 3.0 × 10²⁴ m.
Show the worked solution
- Convert v = 6.6 × 10⁶ m/s.
- Divide 6.6 × 10⁶ by 3.0 × 10²⁴.
- Cancel the metre units.
Answer 2.2 × 10⁻¹⁸ s⁻¹.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
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 v = 4.4 × 10⁶ m/s and d = 2.0 × 10²⁴ m to calculate H₀.
Give me a hint
Divide coefficients and subtract exponents using SI units.
Compare my reasoning
- H₀ = v/d.
- 4.4/2.0 = 2.2 and 10⁶/10²⁴ = 10⁻¹⁸.
- Use the remaining s⁻¹ unit.
2.2 × 10⁻¹⁸ s⁻¹.
Look for these in your work
- Show v/d.
- Give per-second units.
Use H₀ = 2.2 × 10⁻¹⁸ s⁻¹ and d = 5.0 × 10²⁴ m to calculate v.
Give me a hint
Multiply the distance by the constant rather than dividing it.
Compare my reasoning
- Rearrange v = H₀d.
- Calculate 2.2 × 5.0 × 10⁶.
- Normalise the result.
1.1 × 10⁷ m/s.
Look for these in your work
- Use the correct rearrangement.
- Keep the speed unit.
Use H₀ = 2.2 × 10⁻¹⁸ s⁻¹ and v = 8800 km/s to find d.
Give me a hint
Convert speed before dividing by the per-second constant.
Compare my reasoning
- v = 8.8 × 10⁶ m/s.
- d = v/H₀.
- Calculate (8.8/2.2) × 10²⁴.
4.0 × 10²⁴ m.
Look for these in your work
- Convert km/s first.
- Return metres rather than seconds.
Common mistakes
- H₀ is per second, not a speed.
- Use a matching galaxy pair.
- Convert km/s before combining with metres.
What can you explain now?
What is the SI unit of v/d?
Compare with the explanation
s⁻¹
Dividing metres per second by metres leaves per second.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Rearrange H₀ = v/d to find the ratio, speed or distance.
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.