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# Chapter 10 - Systems of Particles and Conservation of Momentum

## Momentum

We can define momentum as the mass times the velocity

$\vec{p}=m\vec{v}$

Momentum has units $\mathrm{kg\,ms^{-1}}$

Unlike kinetic energy momentum is a vector quantity.

The most fundamental aspect of momentum is it's connection to force. We can express Newton's second law as

$\Sigma \vec{F} = \frac{d\vec{p}}{dt}=m\frac{d\vec{v}}{dt}=m\vec{a}$

Over a change in momentum $\Delta p$ from $mv_{1}$ to $mv_{2}$ in a time interval $\Delta t$ during which the instanteous force may vary the average force can also be a useful quantity to consider

$F_{ave}=\frac{\Delta p}{\Delta t}=\frac{mv_{2}-mv_{1}}{\Delta t}$

## Conservation of Momentum

In a collision in which no external force acts the total momentum of the system is a conserved quantity. This can be seen to be a consequence of Newton's third law.

Consider a collision between two objects

$m_{A}\vec{v}_{A}+m_{B}\vec{v}_{B}=m_{A}\vec{v'}_{A}+m_{B}\vec{v'}_{B}$

The force exerted by object A on object B must be

$\vec{F}_{AB}=\frac{d\vec{p}_{B}}{dt}$

And the force exerted by object B on object A must be

$\vec{F}_{BA}=\frac{d\vec{p}_{A}}{dt}$

But we know from Newton's Third Law that

$\vec{F}_{AB}=-\vec{F}_{BA}$

so that

$\frac{d\vec{p}_{A}}{dt}+\frac{d\vec{p}_{B}}{dt}=0$

## Conservation of momentum for many objects

More generally, the total momentum of a system

$\vec{P}=\Sigma_{i} p_{i}=\Sigma_{i} m_{i}v_{i}$

The change of this quantity with time

$\frac{d\vec{P}}{dt}=\Sigma_{i}\frac{d\vec{p_{i}}}{dt}=\Sigma_{i}F_{i}$

$F_{i}$ represents the net force on the $i$th body. All internal force have equal and opposite reaction force on some other body within the system (Newton's 3rd Law) and cancel out in the sum.

So

$\frac{d\vec{P}}{dt}=\Sigma_{i}F_{ext}$

In the absence of an external force the total momentum of a system is constant.

## Rockets

The key to rocket propulsion is that the momentum of the expelled gas is equal and opposite to the forward momentum of the rocket. This does not rely on any interaction with the external atmosphere.