Question #b123c

Apr 3, 2017

See below

Explanation:

If they have different formulae they are not the same:

• Vector quantity momentum is $m a t h b f p = m m a t h b f v$. 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 is conserved, for example, in elastic collisions; but it is not always conserved.

In modern physics, they are often combined as:

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

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

• $p \propto v$

• $T \propto {v}^{\textcolor{red}{2}}$

and

• $T \propto {p}^{2}$

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

$\frac{\mathrm{dT}}{\mathrm{dx}} = m v \frac{\mathrm{dv}}{\mathrm{dx}} = m a$