Fermi estimate, mass of the bullet

Problem

Fermi estimate: mass of the bullet. A $1\;\text{kg}$ pistol recoils at about $3\;\text{m/s}$ when it fires a $9\;\text{mm}$ calibre bullet at a muzzle velocity of $350\;\text{m/s}$. Use conservation of $\vv{p}$ to estimate the mass of the bullet and compare it with real values.

Solution

Before firing, the pistol + bullet system is at rest, so the total momentum is zero. Firing is like a small explosion: the forces are internal, so total $\vv{p}$ is conserved and remains zero. $p_\text{before} = 0 = p_\text{after} = m_p v_p - M_g v_g$ where $m_p$, $v_p$ are the mass and velocity of the bullet and $M_g = 1\;\text{kg}$, $v_g = 3\;\text{m/s}$ are those of the recoil (opposite direction).

Solving for the mass of the bullet: $m_p = \frac{M_g\, v_g}{v_p} = \frac{1 \cdot 3}{350} \approx 8{,}6\cdot 10^{-3}\;\mathrm{kg} = 8{,}6\;\mathrm{g}$ A real $9\;\text{mm}$ bullet typically weighs $7$–$8\;\text{g}$: the estimate is entirely consistent. Result: $\ev{m_p \approx 8{,}6\;\mathrm{g}}$

Links

Topics: Quantità di moto e urti Concepts: Conservazione della quantità di moto · Esplosione Skills: Stime di Fermi Objects: Proiettile