Relativistic Mechanics: Taylor series centered at 0 calculus problem.

Classical Mechanics Level 2

One of the slightly less well known of Einstein's special relativity formulas is the one for mass increase for velocity:

$\text{mass}=\frac{\text{mass}_\text{rest}}{\sqrt{1-(\frac{\text{velocity} }{\text{velocityOfLight}})^2}}$

Rewritten with shorter variable names:

$\text{M}=\frac{m}{\sqrt{1-(\frac{v}{c})^2}}$

$v$ and $c$ have to be expressed in the same units, e.g., $c\ \text{is}\ 299792458 \text{ m/s}, \sim 186282.397 \text{ miles/s}, \sim 67061662.9 \text{ mph}$ or $\sim 1802617499785.25 \text{ furlongs/fortnight}$ .

We will also be using another of Einstein's special relativity formulas: $e=m c^2$ . by using them together.

$e=\frac{m c^2}{\sqrt{1-(\frac{v}{c})^2}}$

Now compute for the equation immediately above the first two terms of the Taylor series centered at $v=0$ . For this problem, those are the only terms you will need.

The question: What do the first two terms represent?

The qualification of " as given in high school general science " is that relativity is not generally taught at the high school level and therefore relativistic corrections are not generally taught. The $e=m c^2$ formula is briefly mentioned in such classes and should be considered when answering.

This is a wrong answer; do not use it. rest mass as energy and kinetic energy as given in high school general science weight and power as given in high school general science rest mass as energy and potential energy as given in high school general science rest mass as inertial mass and kinetic energy as given in high school general science rest mass as inertial mass and potential energy as given in high school general science

1 solution

A Former Brilliant Member
Feb 14, 2019

The first three terms (one more than actually needed) is $c^2 m+\frac{m v^2}{2}+\frac{3 m v^4}{8 c^2}+O\left(v^5\right)$ . Remember that $m$ is the original rest mass . There the first term, $m c^2$ , is the second Einstein formula mentioned and therefore is rest mass as energy . The second term, $\frac{m v^2}{2}$ is the formula general taught for kinetic energy . This explains from where that division by 2 comes. But since as is common neither the teachers nor the students have had calculus, the reason for the division by 2 was not given. The third and remaining terms are sometimes referred to as "relativistic corrections." These corrections are also in energy units (do the dimensional analysis). The corrections are extremely small. At walking speeds, they are in a $10^{-15}$ range.

Relativistic Mechanics: Taylor series centered at 0 calculus problem.

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