Chapter 37: Problem 9
A spacecraft of the Trade Federation flies past the planet Coruscant at a speed of 0.600c. A scientist on Coruscant measures the length of the moving spacecraft to be 74.0 m. The spacecraft later lands on Coruscant, and the same scientist measures the length of the now stationary spacecraft. What value does she get?
Short Answer
Step by step solution
Understand the Lorentz Contraction
Identify the Given Values
Calculate the Proper Length
Compute the Denominator Term
Solve for the Proper Length \( L_0 \)
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Special Relativity
A critical outcome of this theory is the realization that time and space are interconnected into a single entity called spacetime. This means that time can slow down or speed up depending on an object's velocity relative to the observer, and similarly, distances can appear contracted.
Key concepts in Special Relativity include:
- The constancy of the speed of light: The speed of light in a vacuum is the same for all observers, no matter their velocity.
- Time dilation: Clock measurements can differ for observers moving relative to each other.
- Length contraction: Physical lengths appear shorter in the direction of motion when viewed from different inertial frames.
Proper Length
This is because when the object is not moving relative to the observer, the effects of relativistic movement, such as length contraction, do not apply.
For example, if a spacecraft is at rest relative to a planet, and measures 100 meters from front to back according to an onboard observer, that 100 meters is its Proper Length.
In the context of relativity, finding the Proper Length helps us understand how motion affects distances. When an object moves with a significant fraction of the speed of light relative to an observer, the observed length will be less than the Proper Length.
Calculating Proper Length is important for comparing how measurements of the same object can differ between different frames of reference, as demonstrated in the Lorentz contraction formula.
Length Contraction
This contraction is only detectable at relativistic speeds, which are generally much faster than everyday speeds here on Earth.
The mathematical expression to calculate the contracted length, also known as the Lorentz contraction, is: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] Where:
- **\( L \)** is the contracted length viewed by the observer in motion.
- **\( L_0 \)** is the Proper Length, or the length of the object in its rest frame.
- **\( v \)** is the relative velocity of the object and the observer.
- **\( c \)** is the speed of light.