Problem 55 - College Physics

Question

An airplane has a length of 60 \(\mathrm{m}\) when measured at rest. When the airplane is moving at 180 \(\mathrm{m} / \mathrm{s}(400 \mathrm{mph})\) in the alternate uni- verse, how long would it appear to be to a stationary observer? A. 24 \(\mathrm{m}\) B. 36 \(\mathrm{m}\) C. 48 \(\mathrm{m}\) D. 60 \(\mathrm{m}\) E. 75 \(\mathrm{m}\)

Step-by-Step Solution

Answer

The apparent length is 60 meters.

Step 1: Understand the Concept

This problem involves the concept of length contraction in the theory of relativity. A moving object appears shorter in the direction of motion to a stationary observer.

Step 2: Identify the Given Values

We know the rest length of the airplane, \( L_0 = 60 \) m, and its velocity, \( v = 180 \) m/s. We assume the speed of light, \( c \), to be much larger than the velocity, which simplifies the calculations.

Step 3: Recall the Length Contraction Formula

The formula for length contraction is: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]Here, \( L \) is the contracted length seen by the observer.

Step 4: Simplify the Formula

Given that \( v = 180 \) m/s is much smaller than \( c \), the term \( \frac{v^2}{c^2} \) becomes tiny, approaching zero, meaning \( \sqrt{1 - \frac{v^2}{c^2}} \approx 1 \). Thus, \( L = L_0 \).

Step 5: Determine the Apparent Length

Since the relative velocity is negligible compared to the speed of light, the contracted length \( L \approx L_0 = 60 \) m.

Key Concepts

Theory of Relativity

The theory of relativity, introduced by Albert Einstein, revolutionized our understanding of space and time. One of its core ideas is that the laws of physics are the same for all observers, regardless of their relative motion. This theory is divided into two parts: special relativity and general relativity.
Special relativity, which is relevant to our discussion about length contraction, deals with objects moving at constant speeds, particularly those approaching the speed of light. It tells us that time and space are not absolute but relative, depending on the observer’s state of motion. This relativity principle leads to various non-intuitive effects, such as time dilation and length contraction.
These effects become significant only when the objects move at speeds comparable to the speed of light. That's why, in everyday experiences, we don't notice these effects.

Speed of Light

The speed of light in a vacuum is a fundamental constant of nature, typically denoted as \(c\), equating to approximately \(3 imes 10^8\) m/s. In the context of special relativity, the speed of light holds a unique place as it is considered the ultimate speed limit for any form of matter or information in the universe.
One of the remarkable aspects of light's speed is that it remains constant, irrespective of the observer's motion or the source of the light. This constancy challenges our everyday experiences with speed, where relative motion typically affects perceived speeds.

• Light speed is not affected by the relative motion of the source and observer.
• It is the anchor for understanding relativistic effects like time and space transformations.

These properties are crucial for understanding relativistic effects such as those seen in length contraction, where objects seem to contract only at speeds approaching this cosmic speed limit.

Rest Length

Rest length, often symbolized by \(L_0\), is the length of an object measured in the frame in which the object is at rest. For instance, consider a stationary ruler; the length you measure directly is its rest length.
In the context of relativity, the concept of rest length is essential because it serves as the baseline measurement against which we compare the contracted length observed by a moving observer. When an object like an airplane is moving relative to an observer, its length appears shorter; however, its rest length remains unchanged.
Key points to remember about rest length:

• It is the true length of an object, unaffected by motion.
• Acts as a reference for calculating relativistic lengths using the length contraction formula.

Velocity

Velocity, in the realm of physics, is a vector quantity that refers to the speed of an object in a given direction. In the study of relativity, velocity becomes even more intriguing as it directly influences relativistic effects like length contraction and time dilation.
The importance of velocity in understanding relativity comes from the fact that these effects become significant at high velocities, particularly those approaching the speed of light. For our airplane example moving at 180 m/s, the velocity is significant in ordinary terms but negligible in the relativistic context. That's why its length does not appear contracted to the stationary observer.

• Velocity determines the magnitude of relativistic effects.
• Non-relativistic speeds (much smaller than light speed) show minimal length contraction.