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Question #b123c

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Apr 3, 2017

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Explanation:

If they have different formulae they are not the same:

Vector quantity momentum is $p = m v$ $mathbf p = m mathbf v$ . Momentum is always conserved in consequence of Newton's 2nd and 3rd Laws.
Scalar quantity Kinetic Energy is $T = \frac{1}{2} m v^{2}$ $T = 1/2 m v^2$ . T is conserved, for example, in elastic collisions; but it is not always conserved.

In modern physics, they are often combined as:

$T = \frac{p^{2}}{2 m}$ $T = p^2/(2m)$

You are right in one sense though. For constant mass, they are both functions of velocity; but:

$p \propto v$ $p propto v$
$T \propto v^{2}$ $T propto v^color(red)(2)$

and

$T \propto p^{2}$ $T propto p^2$

A more interesting relationship might be, where $x$ $x$ is displacement:

$\frac{d T}{d x} = m v \frac{d v}{d x} = m a$ $(dT)/dx = mv (dv)/dx = ma$

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