Momentum

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Momentum is the measure of 'how' much motion a body posses and is of prime importance to Mechanics. Mathematically if we have a body of mass $m$ moving with velocity $\vec{v}$ then it's momentum $\vec{p}$ is classically defined as

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

Relativistically, momentum is defined as $\vec{p} = \gamma m\vec{v}$ , where $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ is the Lorentz factor dependent on the magnitude of the velocity of the object ( $v$ ), and the speed of light ( $c$ ). For speeds less than about $.1c$ , this definition is approximately equivalent to the classical definition.

The intergalactic speed limit of $c$ follows directly from this definition of momentum, because as the velocity of an object with positive rest mass approaches $c$ , it's momentum must approach $\infty$ , so the force per unit time (i.e. inpulse, or change in momentum) needed to increase the speed further would also approach infinity.

The central importance of momentum is due Newton's second law which states or rather defines Force $\vec{F}$ acting on a body to be equal to it's rate of change of momentum. That is if $\vec{F}$ is the net force acting on a body then we have

$\boxed{\vec{F}=\frac{d \vec{p}}{dt}}$

This means if a body moves such that it's momentum vector remains invariant with time then we can conclude that there is no net Force acting on it.

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