Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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

Expert verified
The proper length of the spacecraft is 92.5 m.

Step by step solution

01

Understand the Lorentz Contraction

The measured length of a moving object contracts in the direction of motion due to special relativity. This phenomenon is known as Lorentz contraction. The formula to calculate the contracted length \( L \) is \( L = L_0 \sqrt{1 - v^2/c^2} \), where \( L_0 \) is the proper length, \( v \) is the velocity of the object, and \( c \) is the speed of light.
02

Identify the Given Values

We are given the velocity \( v = 0.600c \) and the contracted length \( L = 74.0 \text{ m} \). We need to find the proper length \( L_0 \) of the spacecraft when it is at rest.
03

Calculate the Proper Length

Rearrange the Lorentz contraction formula to solve for the proper length \( L_0 \). The formula becomes \( L_0 = \frac{L}{\sqrt{1 - v^2/c^2}} \). Substitute the known values: \( L = 74.0 \text{ m} \), \( v = 0.600c \), and \( c = 1 \) in terms of \( c \).
04

Compute the Denominator Term

Calculate \( 1 - v^2/c^2 \). Since \( v = 0.600c \), we have \( v^2 = (0.600c)^2 = 0.36c^2 \), so \( 1 - v^2/c^2 = 1 - 0.36 = 0.64 \).
05

Solve for the Proper Length \( L_0 \)

Using the adjusted formula \( L_0 = \frac{L}{\sqrt{0.64}} \), substitute \( L = 74.0 \text{ m} \) to find \( L_0 = \frac{74.0}{\sqrt{0.64}} \approx 74.0 \times \frac{5}{4} = 92.5 \text{ m} \).

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
Special Relativity is a fundamental theory in physics developed by Albert Einstein. It describes the behavior of objects moving at speeds close to the speed of light, known as relativistic speeds. One of the key ideas of this theory is that the laws of physics are the same for all observers, regardless of their relative motion.
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.
Overall, Special Relativity introduces a novel way of understanding motion at high speeds, breaking away from classical Newtonian mechanics.
Proper Length
The concept of Proper Length is central in understanding how objects behave under Special Relativity. Proper Length is the length of an object as measured in the object's own rest frame, meaning it's the longest length one can measure for that object.
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
Length Contraction is an intriguing phenomenon predicted by Einstein's theory of Special Relativity. When an object moves at a significant fraction of the speed of light relative to an observer, its length parallel to the direction of motion appears shorter to that observer.
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.
Using this formula, one can determine how long an object appears to an observer witnessing it in motion. As speed increases closer to the speed of light, the observed length decreases, reaching much larger contraction as speeds approach \( c \). This concept was key to solving the exercise involving the spacecraft's observed length on the planet Coruscant.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.150 s. The first officer on the spacecraft measures that the searchlight is on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light \(c\)?

An enemy spaceship is moving toward your starfighter with a speed, as measured in your frame, of 0.400c. The enemy ship fires a missile toward you at a speed of 0.700c relative to the enemy ship (Fig. E37.18). (a) What is the speed of the missile relative to you? Express your answer in terms of the speed of light. (b) If you measure that the enemy ship is 8.00 * 106 km away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you?

The positive muon (\(\mu^+\)), an unstable particle, lives on average 2.20 \(\times\) 10\(^{-6}\) s (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of 0.900c, what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

(a) How fast must you be approaching a red traffic (\(\lambda=\) 675 nm) for it to appear yellow (\(\lambda=\) 575 nm)? Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is $1.00 for each kilometer per hour that your speed exceeds the posted limit of 90 km/h.

(a) At what speed is the momentum of a particle twice as great as the result obtained from the nonrelativistic expression \(mv\)? Express your answer in terms of the speed of light. (b) A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free