Problem — Fermi estimate: g on Mars

Problem

Fermi estimate: $g$ on Mars from observation. A stone falls from $1{,}0\;\text{m}$ on Mars and reaches the ground in about $0{,}74\;\text{s}$. Estimate $g$ on Mars and compare it with Earth’s $g$.

Solution

Model. The stone falls from rest: the free-fall law applies, linking height, time and gravitational acceleration:

$h = \frac{1}{2}\,g_M\,t^2$

Solving for $g_M$. We isolate the Martian gravitational acceleration:

$g_M = \frac{2\,h}{t^2} = \frac{2\cdot 1{,}0}{0{,}74^2} = \frac{2}{0{,}5476}\;\text{m/s}^2$

$\ev{g_M \approx 3{,}7\;\mathrm{m/s}^2}$

Comparison. On Earth $g \approx 9{,}81\;\text{m/s}^2$, so Mars’s gravity is about $38\%$ of Earth’s (a little over a third). On Mars the same stone weighs much less and falls much more slowly: a simple stopwatch and a metre rule are enough to measure its gravity, with no need to know the planet’s mass.

Links

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