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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 \( L_0 = \frac{74.0}{\sqrt{0.64}} \approx 74.0 \times \frac{5}{4} = 92.5 \text{ m} \).

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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.

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Most popular questions from this chapter

The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed \(V\) relative to the lab frame is $$v = {c \over n} + kV$$ where \(n\) = 1.333 is the index of refraction of water. Fizeau called \(k\) the dragging coefficient and obtained an experimental value of \(k\) = 0.44. What value of \(k\) do you calculate from relativistic transformations?

A spacecraft flies away from the earth with a speed of 4.80 \(\times\) 10\(^6\) m/s relative to the earth and then returns at the same speed. The spacecraft carries an atomic clock that has been carefully synchronized with an identical clock that remains at rest on earth. The spacecraft returns to its starting point 365 days (1 year) later, as measured by the clock that remained on earth. What is the difference in the elapsed times on the two clocks, measured in hours? Which clock, the one in the spacecraft or the one on earth, shows the shorter elapsed time?

Everyday Time Dilation. Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 250 m/s and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 4.00 h. By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time? (\(Hint\): Since \(u \ll c\), you can simplify \(\sqrt{1 - u^2/c^2}\) by a binomial expansion.)

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist at rest on the earth's surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle's frame. (c) In the particle's frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

The negative pion (\(\pi^-\)) is an unstable particle with an average lifetime of 2.60 \(\times\) 10\(^{-8}\)s (measured in the rest frame of the pion). (a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be 4.20 \(\times\) 10\(^{-7}\) s. Calculate the speed of the pion expressed as a fraction of c. (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

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