# 3.3.1 Principle of Conservation of Momentum

If forces must come in equal but opposite pairs, then the impulses and momentum changes that they produce must also be equal but opposite. If we sum up all the momentum changes, what do we get? Zero!

This insight is in agreement with the Principle of Conservation of Momentum: since internal forces always come in action-reaction pairs (because N3L), the total momentum of a system must be conserved, unless acted upon by an external force.

Consider an 80-kg astronaut floating in deep space holding a 2-kg spanner in his hand. He then throws the spanner rightward, exerting a 200 N force on the spanner for 0.10 s before the spanner leaves his hand.

In other words, the astronaut exerted a rightward impulse of $200\text{ N }\times \text{ }0.10\text{ s }=20\text{ N s}$ on the spanner. By N3L, the spanner must have exerted a leftward impulse of 20 N s on the astronaut.

For the spanner,

\begin{aligned}(J&=m\Delta v)\\20&=2.0(\Delta v)\\\Delta v&=10\text{ m }{{\text{s}}^{{-1}}}\end{aligned}

So the final velocity of the spanner is 10 m s-1 rightward.

For the astronaut,

\begin{aligned}(J&=m\Delta v)\\-20&=80(\Delta v)\\\Delta v&=-0.25\text{ m }{{\text{s}}^{{-1}}}\end{aligned}

So the final velocity of the astronaut is 0.25 m s-1 leftward.

The magnitude of momentum change for both the astronaut and the spanner are the same. But the velocity change of the astronaut is 1/40 that of the spanner, since its mass is 40 times as large.

Now what about the total momentum? Before the throw, everything was at rest so the total initial momentum was$\displaystyle \sum{{{{p}_{i}}=(2.0)(0)+(80)(0)=0}}$. After the throw, the total momentum is$\displaystyle \sum{{{{p}_{f}}=(2.0)(10)+(80)(-0.25)=0}}$.

Obviously, as individual masses, the momentum of the astronaut and the momentum of the spanner have changed. But as an astronaut-plus-spanner system, the total momentum remains unchanged. Without any external force acting on the astronaut-spanner system, the total momentum must be conserved.

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