Question

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

The proper length of the spacecraft is 92.5 m.

Step-by-step solution

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.

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.

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 $c$.

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 $1-v^2/c^2=1-0.36=0.64$.

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

Key concepts

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.