Fermi estimate, kicking a ball

Problem

Fermi estimate: kicking a ball. Estimate the impulse applied by a footballer to a ball in a well-struck kick, knowing that the ball starts from rest and reaches $30\;\text{m/s}$, with mass $0{,}45\;\text{kg}$. Then estimate the average force on the foot assuming a plausible contact time.

Solution

The impulse is the change in momentum of the ball. It starts from rest ($v_\text{in}=0$) and reaches $v_\text{fin}=30\;\text{m/s}$: $J = \Delta p = m\,(v_\text{fin} - v_\text{in}) = 0{,}45 \cdot 30 = 13{,}5\;\mathrm{N\,s}$

The average force requires the contact time. The foot stays in contact with the ball for a small fraction of a second: estimate $\Delta t \sim 10\;\text{ms} = 0{,}010\;\text{s}$. $F_\text{average} = \frac{J}{\Delta t} \approx \frac{13{,}5}{0{,}010} = 1{,}35\cdot 10^3\;\mathrm{N}$ Order of magnitude $\sim 10^3\;\text{N}$: roughly the weight of $130\;\text{kg}$, plausible for a powerful kick over such a short time. Result: $\ev{J \approx 13{,}5\;\mathrm{N\,s},\quad F_\text{average} \sim 1{,}3\cdot 10^3\;\mathrm{N}}$

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