Problem — Fermi estimate: diving speed

Problem

Fermi estimate: diving speed. Estimate the speed of a diver launching from a height of $10\;\text{m}$, neglecting air resistance. Compare it with the speed of a car in the city.

Solution

Model. Neglecting air resistance, the dive is free fall from rest: all the potential energy converts into kinetic energy. The impact speed depends only on the height via the time-independent relation:

$v = \sqrt{2\,g\,h}$

Numerical estimate. With $g \approx 9{,}81\;\text{m/s}^2$ and $h = 10\;\text{m}$:

$v = \sqrt{2\cdot 9{,}81\cdot 10} = \sqrt{196{,}2}\;\text{m/s}$

$\ev{v \approx 14\;\mathrm{m/s} \approx 50\;\mathrm{km/h}}$

Comparison. That’s about $50\;\text{km/h}$: the same speed as a car speeding through the city. This is why a bad dive from $10\;\text{m}$, belly-flopping onto the water, is equivalent to a small road accident. The free-fall-only estimate works well for the first few metres; at much greater heights air resistance would start to matter.

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Topics: Cinematica Concepts: Caduta libera Skills: Stime di Fermi