One-way third law

Problem

Imagine that Newton’s third law broke down: body A exerts a force on B, but B exerts no reaction on A. Explain in five lines why this would violate conservation of momentum.

Solution

The third law (“for every action there is an equal and opposite reaction”) is deeply linked to the conservation of momentum. Let’s see why.

Consider the isolated system made up of the two bodies A and B alone, with no external forces. The total momentum is $\vv{p} = \vv{p}_A + \vv{p}_B$. Its rate of change over time is the sum of the internal forces acting on the two bodies: $\frac{d\vv{p}}{dt} = \vv{F}_{B\to A} + \vv{F}_{A\to B}$

With the third law intact, $\vv{F}_{B\to A} = -\vv{F}_{A\to B}$: the two internal forces cancel each other out and the total momentum remains constant: $\frac{d\vv{p}}{dt} = \vv{F}_{A\to B} - \vv{F}_{A\to B} = 0 \quad\Rightarrow\quad \vv{p} = \text{constant}$

If the third law broke down, A would exert $\vv{F}_{A\to B}$ on B with no reaction whatsoever ($\vv{F}_{B\to A} = 0$). Then: $\frac{d\vv{p}}{dt} = \vv{F}_{A\to B} \neq 0$

The total momentum of an isolated system would grow (or shrink) out of nothing, with no external force. This directly contradicts the principle of conservation of momentum: a closed system could set itself in motion, “pulling itself up by its own bootstraps”, which is never observed in nature.

$\ev{\vv{F}_{B\to A} = -\vv{F}_{A\to B} \iff \vv{p}_{\text{tot}} = \text{constant}}$

Links

Concepts: Newton’s third law · Conservation of momentum Skills: Analysis of limiting cases and science-fiction physics