If relativistic effects are to be less than 6% then gamma must be less than 1.06. At what...

Question:

If relativistic effects are to be less than 6% then gamma must be less than 1.06. At what relative velocity is gamma = 1.06?

Lorentz Factor:

The Lorentz factor is a correction factor used for relativity calculations. Without this factor, the calculations for relativistic motion simply break down and we can get speeds that exceed the speed of light which cannot be.

Answer and Explanation:


Given:

  • {eq}\displaystyle \gamma = 1.06 {/eq} is the Lorentz factor


In general, we can write the Lorentz factor as:

{eq}\displaystyle \gamma = \frac{1}{\sqrt{1-\dfrac{v^2}{c^2}}} {/eq}

So here we square both sides:

{eq}\displaystyle \gamma^2 = \frac{1}{1-\dfrac{v^2}{c^2}} {/eq}

We take the reciprocals of both sides:

{eq}\displaystyle 1 - \frac{v^2}{c^2} = \frac{1}{\gamma^2} {/eq}

We transpose:

{eq}\displaystyle \frac{v^2}{c^2} = 1 - \frac{1}{\gamma^2} {/eq}

We isolate v here as:

{eq}\displaystyle v = \sqrt{1 - \frac{1}{\gamma^2}} c {/eq}

We substitute our Lorentz factor:

{eq}\displaystyle v = \sqrt{1 - \frac{1}{1.06^2}} c {/eq}

Relative to the speed of light, we thus get:

{eq}\displaystyle \boxed{v = 0.33c} {/eq}


Learn more about this topic:

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Theory of Relativity: Definition & Example

from Remedial Algebra I

Chapter 25 / Lesson 1
16K

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