Key ideas
Earth science problems turn a few logarithmic and linear relations into numbers. Earthquake magnitude is logarithmic, seismic wave lag gives distance, and rates of plate motion and radioactive decay give ages.
- Each magnitude unit multiplies the recorded wave amplitude by 10 and the released energy by 10^1.5, about 31.6.
- The delay between P and S waves gives the distance d = Δt × vP·vS ÷ (vP − vS).
- The air temperature falls about 6.5 °C for every kilometer of altitude in the lower atmosphere.
- Crust age is distance divided by spreading rate. A radiometric age is the half-life times the number of halvings, t = t½ × log₂(N₀ ÷ N).
Key equations
amplitude ratio = 10^ΔMenergy ratio = 10^(1.5 ΔM)d = Δt × vP·vS ÷ (vP − vS)T = T₀ − 6.5 × ht = d ÷ vt = t½ × log₂(N₀ ÷ N)
Worked example
A magnitude 7 earthquake against magnitude 5 has 10² = 100 times the amplitude and 10^(1.5 × 2) = 1000 times the energy. With vP = 6 km/s, vS = 3.5 km/s and a 20 s lag, d = 20 × (6 × 3.5) ÷ 2.5 = 168 km. A 15 °C sea level at 3 km altitude gives 15 − 6.5 × 3 = −4.5 °C. Crust 100 km from a ridge spreading at 2 cm per year is 10⁷ cm ÷ 2 = 5 × 10⁶ years, or 5 million years, old. A sample with 25 % of its carbon-14 left has gone through 2 half-lives, so 2 × 5730 = 11 460 years.
Common mistakes
- Treating magnitude as linear, so that a 2-unit difference becomes 2 times instead of 100 times in amplitude.
- Mixing km and cm when dividing distance by spreading rate. Convert both to the same unit first.
- Using the fraction remaining as the number of half-lives. 25 % is two halvings, not 0.25.
Practice
Test yourself with practice questions on this topic: Earthquakes, plates and dating