Chapter 37: Problem 28
As you have seen, relativistic calculations usually involve the quantity \(\gamma\). When \(\gamma\) is appreciably greater than 1, we must use relativistic formulas instead of Newtonian ones. For what speed \(v\) (in terms of \(c\)) is the value of \(\gamma\) (a) 1.0% greater than 1; (b) 10% greater than 1; (c) 100% greater than 1?
Short Answer
Step by step solution
Understand gamma
Set up the equation for (a)
Solve equation for (a)
Compute (a) result
Set up and solve equation for (b)
Compute (b) result
Set up and solve equation for (c)
Compute (c) result
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Special Relativity
It introduces a new framework where the laws of physics remain constant for observers moving at constant speeds relative to one another.
A key insight is that the speed of light is constant and acts as a cosmic speed limit that nothing can surpass. This leads to intriguing consequences like time dilation and length contraction, concepts which become significant at high speeds, approaching the speed of light.
The Lorentz factor, \( \gamma \), is central to these phenomena and adjusts the equations governing time and space, making relativistic effects measurable.
Relativistic Velocity
When objects move at speeds close to that of light, their apparent velocity changes are linked with time dilation and length contraction.
The formula for relativistic addition of velocities ensures that the resulting speed never exceeds the speed of light, maintaining Einstein's postulate.
Understanding relativistic velocity is crucial in high-energy physics and astronomy where objects frequently move at speeds significant compared to the speed of light.
Lorentz Transformation
They mathematically formalize the concepts introduced by special relativity, adjusting measurements between observers in different inertial frames.
These transformations ensure that the speed of light remains constant and implement corrections for time dilation and length contraction.
The transformations have far-reaching implications, demonstrating that measurements of time and length are relative to the motion of the observer, shaking the foundation of classical mechanics.
Speed of Light
It serves as a fundamental constant in nature and a cornerstone of Einstein's theory of special relativity.
This speed limit is not just for light, but for all information and matter, playing a crucial role in the relativistic equations and concepts.
By establishing an upper boundary for speed, it reshapes how we perceive motion and energy, leading to a deeper understanding of the universe's operations and constraints.