Question

Suppose your handheld calculator will show six places beyond the decimal point. At what minimum speed would an object have to be traveling so that gamma can be seen to be different from 1 on your calculator display? That is, how fast should an object travel so that $\gamma=1.000001 ?[$Hint: Use the binomial approximation. $]$

Name: Problem 7, Chapter 26: Physics
Uploaded: 2021-04-05T06:12:26-08:00
Duration: 1 min 29 s
Channel: Narayan Hari
Description: Suppose your handheld calculator will show six places beyond the decimal point. At what minimum speed would an object have to be traveling so that gamma can be seen to be different from 1 on your calculator display? That is, how fast should an object travel so that γ=1.000001 ?[Hint: Use the binomial approximation. ]

   Suppose your handheld calculator will show six places beyond the decimal point. At what minimum speed would an object have to be traveling so that gamma can be seen to be different from 1 on your calculator display? That is, how fast should an object travel so that $\gamma=1.000001 ?[$Hint: Use the binomial approximation. $]$

Physics

Robert Coleman… 5th Edition

Chapter 26, Problem 7

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Solved by Expert Narayan Hari

04/05/2021

Step 1

Step 1: We start with the Lorentz factor (gamma) equation which is given by: \[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\] Show more…

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Suppose your handheld calculator will show six places beyond the decimal point. At what minimum speed would an object have to be traveling so that gamma can be seen to be different from 1 on your calculator display? That is, how fast should an object travel so that $\gamma=1.000001 ?[$Hint: Use the binomial approximation. $]$

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Key Concepts

Lorentz Factor

In special relativity, the Lorentz factor, denoted as gamma (?), is defined as 1/sqrt(1 - v²/c²) and plays a fundamental role in quantifying time dilation, length contraction, and relativistic mass. It expresses how measurements of time and space change for observers in different inertial frames as an object’s speed approaches the speed of light.

Binomial Approximation

The binomial approximation is a mathematical method used to simplify expressions involving powers, such as approximating (1 - x)^(-1/2) as 1 + (x/2) when x is much less than 1. In the context of special relativity, it provides an efficient way to estimate small deviations of the Lorentz factor from unity for speeds that are much lower than the speed of light.

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