Jumping down from a stool

Problem

Fermi estimate: jumping down from a stool. Estimate the average force the floor exerts on the feet when you jump down from a stool $40\;\text{cm}$ high and land, cushioning the fall in $0{,}25\;\text{s}$. Person of $70\;\text{kg}$.

Solution

Impact velocity (fall from $h = 0{,}40\;\text{m}$): $v = \sqrt{2 g h} = \sqrt{2\cdot 9{,}81\cdot 0{,}40} \approx 2{,}8\;\text{m/s}$

Deceleration during cushioning ($\Delta t = 0{,}25\;\text{s}$): $a = \frac{v}{\Delta t} = \frac{2{,}8}{0{,}25} \approx 11{,}2\;\text{m/s}^2$

Average force on the floor (must support the weight and brake the fall, $F = m(g + a)$): $F = 70\cdot(9{,}81 + 11{,}2) \approx 1{,}5\cdot 10^{3}\;\text{N}$

$\ev{F \approx 1{,}5\cdot 10^{3}\;\text{N} \approx 2{,}1\ \text{times the weight}}$ Cushioning for longer (increasing $\Delta t$) would reduce this force.

Links

Topics: Quantità di moto e urti Concepts: Impulso · Seconda legge di Newton Skills: Stime di Fermi