Water bottle rocket

Problem

Fermi estimate: bottle rocket. A plastic bottle turned into a compressed-air rocket has a mass of $200\;\text{g}$ and expels $100\;\text{g}$ of water at $20\;\text{m/s}$. Estimate the initial velocity it reaches.

Solution

Conservation of momentum (system initially at rest). The expelled water carries away momentum $m_a v_a$ downward; the bottle recoils upward: $m_b\,v_b = m_a\,v_a \;\Rightarrow\; v_b = \frac{m_a\,v_a}{m_b} = \frac{0{,}100\cdot 20}{0{,}200}$ $\ev{v_b \sim 10\;\text{m/s upward}}$ This is an estimate: the jet is not instantaneous and the mass of the bottle changes as it expels the water, but the order of magnitude is correct.

Links

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