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In the theory of relativity, the mass of a particle with speed $ v $ is

$$ m = f(v) = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} $$

where $ m_0 $ is the rest mass of the particle and $ c $ is the speed of light in a vacuum. Find the inverse function of $ f $ and explain its meaning.

$v=f^{-1}(m)=c \sqrt{1-\frac{m_{0}^{2}}{m^{2}}}$

03:19

Jeffrey P.

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 5

Inverse Functions and Logarithms

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

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the function we're looking at here shows Mass as a function of speed. The inverse would show speed as a function of mass. So what we want to do is take this equation and isolate V. Isolate the speed so we'll go through the algebraic process to do that. So the first thing we could do is multiply both sides by the square root. So we have m times. The square root of one minus B squared over C squared equals am not. Then we'll divide both sides by m, and we have the square root of one minus B squared over C squared equals m divided by AM not divided by em. Then we'll square both sides and we have one minus B squared over C squared equals m, not squared over m squared. Then we'll subtract one from both sides and we have the opposite of V squared over C squared equals m not squared over m squared minus one. Then we'll multiply both sides by negative C squared to get B squared by itself to a B squared equals negative C squared times m squared and not squared over m squared minus one. Then we'll distribute the negative. Sign through the parentheses and we have V squared equals C squared times one minus AM not squared over m squared. Finally, we'll square root to get V by itself and we have V equals the square root of C squared times one minus m not squared over m squared. We can simplify that by taking the square root of C squared and we have equal C times the square root of one minus am not squared over m squared. And our interpretation of that is that we now have speed as a function of mass rather than mass as a function of speed.

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