Mass that halves

Problem

Thought experiment: a mass that halves. A $2\;\text{kg}$ ball travels at $10\;\text{m/s}$ and, purely hypothetically, instantly halves its mass while keeping the same velocity. What would happen to $\vv{p}$ and $E_\text{kin}$? Why can this not happen in nature?

Solution

Initial state: $m_0 = 2\;\text{kg}$, $v = 10\;\text{m/s}$. $p_0 = m_0 v = 2 \cdot 10 = 20\;\mathrm{kg\,m/s}$ $E_{\text{kin},0} = \tfrac{1}{2} m_0 v^2 = \tfrac{1}{2}\cdot 2 \cdot 10^2 = 100\;\mathrm{J}$ If the mass halves to $m_1 = 1\;\text{kg}$ while keeping the same $v$: $p_1 = m_1 v = 1 \cdot 10 = 10\;\mathrm{kg\,m/s}$ $E_{\text{kin},1} = \tfrac{1}{2} m_1 v^2 = \tfrac{1}{2}\cdot 1 \cdot 10^2 = 50\;\mathrm{J}$ So both $\vv{p}$ and $E_\text{kin}$ would halve: both vanish without anything carrying them away. Why it’s impossible: in classical mechanics the mass of an isolated system is conserved. Making half the mass disappear would violate conservation of momentum (where would the missing $\vv{p}$ go?) and of energy. Only relativistic mass-energy conversions or decays can change the mass, and even then they always balance $\vv{p}$ and energy. Result: $\ev{\text{Impossible: mass conserved; } \vv{p} \text{ and } E_\text{kin} \text{ cannot halve on their own}}$

Links

Topics: Momentum and collisions Concepts: Momentum · Kinetic energy · Conservation of momentum Skills: Limiting-case and thought-experiment analysis