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Chapter 35: Problem 24

What is the value of \(\gamma\) for a particle moving at a speed of \(0.8 c ?\)

Short Answer

Expert verified
Answer: The Lorentz factor for a particle moving at a speed of 0.8c is approximately 1.67.

Step by step solution

01

Understand the Lorentz Factor Formula

The Lorentz factor, denoted by \(\gamma\), is a fundamental concept in special relativity. It relates the time, length, and mass of an object moving at relativistic speeds to those measured in its rest frame. The formula to calculate the Lorentz factor is: $$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$ Where: - \(v\) is the speed of the particle - \(c\) is the speed of light
02

Insert the Given Values

We are given that the particle is moving at a speed of \(0.8c\). To find the value of \(\gamma\), we'll substitute the given values into the formula: $$\gamma = \frac{1}{\sqrt{1 - \frac{(0.8c)^2}{c^2}}}$$
03

Simplify the Expression

Now, let's simplify the expression by squaring the speed of the particle, dividing by the square of the speed of light, and subtracting the result from 1: $$\gamma = \frac{1}{\sqrt{1 - \frac{(0.64c^2)}{c^2}}}$$
04

Cancel out the Speed of Light

The \(c^2\) in the numerator and denominator will cancel each other out, which leaves us with: $$\gamma = \frac{1}{\sqrt{1 - 0.64}}$$
05

Solve for Gamma

Finally, we can calculate the value of \(\gamma\) by finding the square root: $$\gamma = \frac{1}{\sqrt{0.36}}$$ $$\gamma = \frac{1}{0.6}$$ $$\gamma \approx 1.67$$ Therefore, the value of \(\gamma\) for a particle moving at a speed of \(0.8c\) is approximately 1.67.

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Most popular questions from this chapter

A famous result in Newtonian dynamics is that if a particle in motion collides elastically with an identical particle at rest, the two particles emerge from the collision on perpendicular trajectories. Does the same hold in the special theory of relativity? Suppose a particle of rest mass \(m\) and total energy \(E\) collides with an identical particle at rest, the same two particles emerging from the collision with new velocities. Are those velocities necessarily perpendicular? Explain.

The most important fact we learned about aether is that: a) It was experimentally proven not to exist. b) Its existence was proven experimentally. c) It transmits light in all directions equally. d) It transmits light faster in longitudinal direction. e) It transmits light slower in longitudinal direction.

Consider a particle moving with a speed less than \(0.5 c\). If the speed of the particle is doubled, by what factor will the momentum increase? a) less than 2 b) equal to 2 c) greater than 2

An electron is accelerated from rest through a potential of \(1.0 \cdot 10^{6} \mathrm{~V}\). What is its final speed?

Prove that in all cases, two sub-light-speed velocities "added" relativistically will always yield a sub-light-speed velocity. Consider motion in one spatial dimension only.
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