Problem 23 - Fundamentals of Physics

Question

A clock moves along an \(x\) axis at a speed of \(0.600 \mathrm{c}\) and reads zero as it passes the origin of the axis. (a) Calculate the clock's Lorentz factor. (b) What time does the clock read as it passcs \(x=180 \mathrm{~m} ?\)

Step-by-Step Solution

Answer

(a) Lorentz factor is 1.25. (b) Clock reads 0.8 microseconds.

Step 1: Identify Given Variables

We are given that the speed of the clock is \(v = 0.600c\), where \(c\) is the speed of light. We also know that the clock reads zero at the origin and we need to find the time when \(x = 180 \, \text{m}\).

Step 2: Calculate Lorentz Factor

The Lorentz factor \(\gamma\) is calculated using the formula: \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]Substitute \(v = 0.600c\):\[ \gamma = \frac{1}{\sqrt{1 - (0.600)^2}} = \frac{1}{\sqrt{1 - 0.3600}} = \frac{1}{\sqrt{0.6400}} = \frac{1}{0.8} = 1.25 \]

Step 3: Use Time Dilation Formula

The time read by the moving clock \(t'\) is related to the time \(t\) measured in the stationary frame by the formula:\[ t' = \frac{t}{\gamma} \]Where \(t\) is the time taken in the stationary frame. First, calculate \(t\):\[ t = \frac{x}{v} = \frac{180 \, \text{m}}{0.600c} \]Substitute \(c = 3 \times 10^8 \, \text{m/s}\):\[ t = \frac{180}{0.600 \times 3 \times 10^8} = \frac{180}{1.8 \times 10^8} = 1 \times 10^{-6} \, \text{s} \]

Step 4: Calculate Time on Moving Clock

Now apply time dilation to find \(t'\):\[ t' = \frac{t}{\gamma} = \frac{1 \times 10^{-6} \, \text{s}}{1.25} = 0.8 \times 10^{-6} \, \text{s} = 0.8 \, \text{us} \]

Key Concepts

Lorentz factor

The Lorentz factor is a crucial element in the theory of Special Relativity. It explains how much time, length, and relativistic mass scale change for an object moving at a significant fraction of the speed of light. To calculate the Lorentz factor, use the formula:\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]Here:

• \( v \) is the velocity of the moving object.
• \( c \) represents the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \).

In the provided example, the clock's speed is given as \( 0.600c \). By substituting this value into the formula, the Lorentz factor becomes:\[ \gamma = \frac{1}{\sqrt{1 - (0.600)^2}} = 1.25 \]This factor indicates how much the time, length, and other quantities will be affected by the motion of the clock compared to a stationary observer.

Time dilation

Time dilation is another fundamental concept in Special Relativity. It describes how time passes at different rates for observers in different frames of reference, especially when moving at relativistic speeds. This can be seen as one of the peculiar effects of traveling close to the speed of light.The formula to calculate time dilation is:\[ t' = \frac{t}{\gamma} \]Where:

• \( t \) is the time interval measured by the stationary observer.
• \( t' \) is the time interval experienced by the moving clock.
• \( \gamma \) is the Lorentz factor.

In our exercise, after calculating the stationary time \( t \), which was \( 1 \times 10^{-6} \, \text{s} \), we found the moving time by dividing it by the Lorentz factor \( 1.25 \). This resulted in:\[ t' = 0.8 \times 10^{-6} \, \text{s} = 0.8 \, \text{us} \]Meaning, the clock moving at \( 0.600c \) would read this shorter time due to time dilation effects.

Speed of light

The speed of light is the ultimate speed limit in the universe and plays a pivotal role in the theory of Special Relativity. It forms the basis for understanding why powerful effects like time dilation and length contraction occur in high-speed travel.In most relative calculations, the speed of light is denoted as \( c \) and has a constant value of approximately \( 3 \times 10^8 \, \text{m/s} \). The constancy of the speed of light means that it remains the same in all inertial frames of reference, regardless of the motion of the light source or observer.This foundational concept was one of Einstein's postulates when he developed Special Relativity. It requires that:

• All observers measure the speed of light as the same value \( c \), no matter how they themselves are moving relative to the light source.
• Time and space coordinates must transform between observers using the Lorentz transformations, which are directly related to the speed of light constraints.

Understanding the speed of light helps us grasp why phenomena like time dilation occur when objects move at speeds close to \( c \). It also shows that no object with mass can reach or exceed this speed, as their mass and energy requirements would become infinite.