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 passes \(x=180 \mathrm{~m} ?\)

Step-by-Step Solution

Answer

(a) \( \gamma = 1.25 \), (b) Clock reads \( 0.8 \times 10^{-6} \, ext{s} \).

Step 1: Understanding the Lorentz Factor

The Lorentz factor, denoted by \( \gamma \), is a key concept in special relativity. It is defined as \( \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \), where \( v \) is the speed of an object and \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \). In this exercise, \( v = 0.600c \).

Step 2: Calculate the Lorentz Factor

Substitute the given speed into the Lorentz factor equation:\[\gamma = \frac{1}{\sqrt{1-\left(0.600\right)^2}}\]Calculating the expression inside the square root gives:\[1-0.600^2 = 1 - 0.36 = 0.64\]Then calculate the entire Lorentz factor:\[\gamma = \frac{1}{\sqrt{0.64}} = \frac{1}{0.8} = 1.25\]

Step 3: Understanding Time Dilation Calculation

In special relativity, the relationship between the time observed in a moving system (\( t' \)) and a stationary system is given by the formula \( t' = \frac{t}{\gamma} \). However, the clock directly gives us \( t' \) when calculating from its own frame.

Step 4: Calculate Time at \(x = 180 \, \text{m}\)

The distance \( x = 180 \, \text{m} \) is the path covered by the clock as observed from the stationary frame. The actual time \( t \) taken in the stationary frame is \( t = \frac{x}{v} = \frac{180 \, ext{m}}{0.600c} \). Convert this: \[ t = \frac{180 \, ext{m}}{0.600 \times 3 \times 10^8 \, \text{m/s}} \approx 1.0 \times 10^{-6} \, ext{s} \]

Step 5: Determine Clock's Reading

The time the clock reads \( t' \) is the real time it experiences, calculated by:\[ t' = \frac{t}{\gamma} = \frac{1.0 \times 10^{-6} \, \text{s}}{1.25} \approx 0.8 \times 10^{-6} \, ext{s} \]

Key Concepts

The Lorentz Factor

The Lorentz factor, represented by the Greek letter \( \gamma \), is an essential concept in the realm of special relativity. This factor is crucial when observing phenomena at speeds approaching the speed of light, \( c \), which is approximately \( 3 \times 10^8 \, \text{m/s} \).

To calculate the Lorentz factor, you use the formula:

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

where \( v \) is the object's velocity.
This factor explains how time, length, and relativistic mass change for an object while moving. For instance, when \( v = 0.600c \), the Lorentz factor becomes:

• \( \gamma = \frac{1}{\sqrt{1-(0.600)^2}} \)

Upon calculating, \( \gamma = 1.25 \).
This means that time will appear to slow down by a factor of 1.25 for the observer in motion compared to one at rest.

Time Dilation

Time dilation is a fascinating consequence of Einstein's theory of special relativity. It describes how time behaves differently for observers moving relative to one another. When you travel at significant fractions of the speed of light, time for you will slow compared to someone at rest – this is called dilated time.

The relationship between the time\( t \) in a stationary frame and \( t' \) in the moving frame is given by:

• \( t' = \frac{t}{\gamma} \)

In this formula, \( \gamma \) is the Lorentz factor.
Let's say a clock is moving at 0.600c and reaches 180 meters in the stationary frame.
In this case:

• The stationary observer calculates \( t \) as \( \frac{180 \, \text{m}}{0.600c} \approx 1.0 \times 10^{-6} \, \text{s} \)
• For the moving clock, \( t' = \frac{1.0 \times 10^{-6} \, \text{s}}{1.25} \approx 0.8 \times 10^{-6} \, \text{s} \)

Thus, the clock reads a shorter time, illustrating how time dilation impacts time perception in different frames of reference.

The Speed of Light

The speed of light, \( c \), is one of the most critical constants in physics, underpinning many principles of both special and general relativity. It is precisely \( 299,792,458 \, \text{m/s} \) and is considered a universal constant.

In special relativity:

• Nothing with mass can reach or exceed it, ensuring that travel to distant galaxies remains solely in the realm of science fiction for now.
• It represents the ultimate speed limit in our universe, a fact that every theory in physics takes into account.

In our exercise, when the clock moves at \( 0.600c \), it means the clock is traveling at 60% of this cosmic speed limit.
Multiple predictions and equations, including those for time dilation and the Lorentz factor, are deeply rooted in this constant, reinforcing the speed of light's pivotal role in understanding the nature of time and space.