Problem 9 - University Physics with Modern Physics

Question

A spacecraft of the Trade Federation flies past the planet Coruscant at a speed of 0.600 \(\mathrm{c}\) . A scientist on Coruscant measures the length of the moving spacecraft to be 74.0 \(\mathrm{m}\) . The spacecraft later lands on Coruscant, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

Step-by-Step Solution

Answer

The proper length of the spacecraft is 92.5 m.

Step 1: Identify the Given Information

We have a moving spacecraft with a relative speed of 0.600 \( c \), where \( c \) is the speed of light. The Lorentz contraction formula relates the length of the object in the frame where it moves to the length in its rest frame.

Step 2: Use the Lorentz Contraction Formula

The formula for length contraction is \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L \) is the contracted length measured in the moving observer's frame (74.0 m), \( L_0 \) is the proper length (length measured in the object's rest frame), and \( v \) is the velocity (0.600 \( c \)).

Step 3: Rearrange the Formula to Solve for Proper Length

We need \( L_0 \), so rearrange the formula: \( L_0 = \frac{L}{\sqrt{1 - \frac{v^2}{c^2}}} \).

Step 4: Plug in the Given Values

Substitute the given values into the formula: \[ L_0 = \frac{74.0}{\sqrt{1 - (0.600)^2}} \].

Step 5: Calculate the Proper Length

First, calculate \( 1 - (0.600)^2 = 1 - 0.36 = 0.64 \). The square root is \( \sqrt{0.64} = 0.8 \). Therefore, \[ L_0 = \frac{74.0}{0.8} = 92.5 \; \text{m} \].

Key Concepts

Special Relativity

The theory of Special Relativity, proposed by Albert Einstein in 1905, revolutionized our understanding of space and time. This theory introduces the idea that the laws of physics are the same for all non-accelerating observers. It also dictates that the speed of light is constant, regardless of the observer's motion. This leads to fascinating phenomena like time dilation and length contraction.

When objects move at speeds close to the speed of light, their lengths and the duration of events are perceived differently by stationary observers compared to those moving with the object. These effects are not noticeable at everyday speeds, which is why we don't experience them in our daily lives. However, they become significant at relativistic speeds approaching the speed of light. Understanding these concepts helps us delve deeper into the fabric of the universe.

Lorentz Contraction Formula

The Lorentz Contraction Formula is a key component in explaining length contraction in Special Relativity. It models how an object's length appears shorter to an observer when the object is moving at high velocities compared to a stationary observer. This formula is expressed as:

• \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \)

Here:

• \( L \) is the observed length in the frame where the object is moving.
• \( L_0 \) is the 'proper length' or the length of the object in its rest frame, where the object is not moving.
• \( v \) is the relative velocity of the object.
• \( c \) is the speed of light.

This formula shows that as the velocity \( v \) approaches \( c \), the observed length \( L \) becomes significantly smaller compared to the proper length \( L_0 \). This is why the spacecraft measured 74 meters while in motion and 92.5 meters when stationary.

Proper Length

Proper Length, in the context of Special Relativity, refers to the length of an object measured in the object's own rest frame. This is a key concept because it represents the maximum length of the object, undistorted by the effects of its motion relative to an observer.

When an object travels at a significant fraction of the speed of light, it undergoes length contraction, meaning its observed length shortens. The great aspect of the proper length is that it remains constant because it is measured when the object is not moving. In our example, the spacecraft's proper length was found to be 92.5 meters. Knowing the proper length allows physicists to predict how an object will appear to move observers at different speeds.

In sum, the difference between the proper length and the contracted length is a beautiful illustration of how motion and speed can distort the measurements of space.