Problem 12 - College Physics

Question

In the year \(2084,\) a spacecraft flies over Moon Station III at a speed of 0.800\(c .\) A scientist on the moon measures the length of the moving spacecraft to be 140 \(\mathrm{m} .\) The spacecraft later lands on the moon, and the same scientist measures the length of the now stationary spacecraft. What value does she get?

Step-by-Step Solution

Answer

The scientist measures the stationary spacecraft to be 233.33 meters long.

Step 1: Identify the problem

The problem involves a spacecraft traveling at a high speed relative to an observer, requiring an application of the concept of length contraction in special relativity.

Step 2: Understand length contraction formula

The formula used for length contraction is \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L \) is the length measured by the observer, \( L_0 \) is the proper length, \( v \) is the velocity of the object, and \( c \) is the speed of light.

Step 3: Assign given values and solve for proper length

We know \( v = 0.800c \), \( L = 140 \, \text{m} \), and \( c \) is the speed of light.Rearrange the formula to find the proper length (\( L_0 \)):\[ L_0 = \frac{L}{\sqrt{1 - \frac{v^2}{c^2}}} \]Substitute the values:\[ L_0 = \frac{140}{\sqrt{1 - (0.800)^2}} \]\[ L_0 = \frac{140}{\sqrt{1 - 0.6400}} \]\[ L_0 = \frac{140}{\sqrt{0.3600}} \]\( L_0 = \frac{140}{0.6} \)\( L_0 = 233.33 \, \text{m} \)

Step 4: Interpretation of results

The proper length \( L_0 \) is the actual length of the spacecraft when it is measured at rest. This means when the spacecraft is stationary, the scientist will measure it to be 233.33 meters long.

Key Concepts

Length Contraction

In the fascinating realm of special relativity, **length contraction** is a pivotal concept. It describes how the measured length of an object changes depending on its velocity relative to an observer.

- According to Einstein's theory, when an object travels close to the speed of light, its length appears shorter from the viewpoint of a stationary observer. - This phenomenon is encapsulated in the length contraction formula: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] where: - \( L \) is the observed length of the moving object, - \( L_0 \) (proper length) is the length of the object when at rest, - \( v \) is the velocity of the object, and - \( c \) is the speed of light.This effect occurs because as objects approach the speed of light, time and space coordinate differently than in the everyday experiences of slow-moving objects. This differential in observable length becomes significant only at high velocities, such as the one mentioned in the scenario where the spacecraft moves at 0.800 times the speed of light.

Proper Length

**Proper length** is a key term in understanding length contraction. It refers to the length of an object as measured in its own rest frame. Unlike the contracted length, which an observer measures for a moving object, the proper length remains unchanged regardless of the object's motion because it's measured where the object isn't moving.

- In the example problem, the spacecraft's proper length, found using rearranged contraction formula, is around 233.33 meters.- This is the true length of the spacecraft when it is not speeding by at relativistic velocities.- To find it, we rearrange the formula for length contraction to solve for \( L_0 \): \[ L_0 = \frac{L}{\sqrt{1 - \frac{v^2}{c^2}}} \]Proper length is important as it represents the object's actual dimensions when observed without the distortion caused by relative motion.

Velocity and Speed of Light

Central to the notion of relativity is how velocity and the speed of light interact. Einstein's theories propose that the speed of light, denoted as \( c \), is a universal constant, unchanged regardless of the motion of the light source or observer. This forms the backdrop for phenomena like length contraction.

- **Velocity**, represented as \( v \) in our formulas, is the speed and direction of a moving object.- In the realm of special relativity, as an object's velocity nears the speed of light, dramatic changes in time and space occur, like the contraction of length.- The speed of light sets a cosmic speed limit, meaning no matter can reach or exceed it.In the presented exercise, the spacecraft travels at 0.800 times the speed of light, illustrating how significant relativistic effects like length contraction become apparent when dealing with high velocities in respect to the speed of light. Understanding this interaction is crucial in grasping the comprehensive effects of special relativity.