Problem 10 - University Physics with Modern Physics

Question

A meter stick moves past you at great speed. Its motion relative to you is parallel to its long axis. If you measure the length of the moving meter stick to be \(1.00 \mathrm{ft}(1 \mathrm{ft}=0.3048 \mathrm{m})-\) for example, by comparing it to a 1 -foot ruler that is at rest relative to you \(-\) at what speed is the meter stick moving relative to you?

Step-by-Step Solution

Answer

The meter stick is moving at approximately \( 2.8563 \times 10^8 \text{ m/s} \).

Step 1: Understand Length Contraction in Special Relativity

According to the theory of special relativity, the observed length of an object moving relative to an observer is shorter than when it is at rest. This phenomenon is known as length contraction and is given by the formula: \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L \) is the length of the object as measured by the observer, \( L_0 \) is the length of the object in its rest frame (proper length), \( v \) is the velocity of the object, and \( c \) is the speed of light.

Step 2: Identify Given Values

From the problem, you have the proper length, \( L_0 = 1 \text{ m} \), and the contracted length measured by you, \( L = 1 \text{ ft} = 0.3048 \text{ m} \). We need to find the speed \( v \) of the meter stick relative to the observer.

Step 3: Rearrange the Length Contraction Formula

Rearrange the length contraction formula to solve for \( v \): \( v = c \sqrt{1 - \left( \frac{L}{L_0} \right)^2} \).

Step 4: Substitute Known Values Into the Equation

Substitute the known values into the equation: \( L = 0.3048 \text{ m} \), \( L_0 = 1 \text{ m} \), and \( c \approx 3 \times 10^8 \text{ m/s} \). The equation becomes \( v = 3 \times 10^8 \sqrt{1 - (0.3048)^2 } \).

Step 5: Calculate the Result

Perform the calculation: First, calculate \( \left( \frac{L}{L_0} \right)^2 = (0.3048)^2 \approx 0.0929 \). Then find \( 1 - 0.0929 = 0.9071 \). Finally, calculate \( \sqrt{0.9071} \approx 0.9521 \). Therefore, \( v \approx 3 \times 10^8 \times 0.9521 \approx 2.8563 \times 10^8 \text{ m/s} \).

Step 6: Conclusion: Express the Result

The speed of the meter stick relative to the observer is approximately \( 2.8563 \times 10^8 \text{ m/s} \).

Key Concepts

Length Contraction

Length contraction is a fascinating concept in the realm of special relativity. It describes how the length of an object moving at high speeds becomes shorter when observed from a stationary point of view. The formula to understand this phenomenon is:

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

Here:

• \( L \) is the contracted length observed.
• \( L_0 \) is the actual length when at rest.
• \( v \) is the speed of the object.
• \( c \) is the speed of light, approximately \( 3 \times 10^8 \text{ m/s} \).

As the speed of the object approaches the speed of light, the length contraction becomes more pronounced. It's important to note that this contraction is only noticeable at speeds close to the speed of light. For everyday speeds, like a moving car or plane, length contraction is not apparent. This concept helps us understand the behavior of particles in accelerators and is crucial in astrophysics.

Lorentz Transformation

The Lorentz Transformation equations are the mathematical tools that allow us to change the coordinates of events in spacetime from one inertial frame to another. They form the backbone of Einstein’s theory of special relativity and ensure that the speed of light is constant in all frames of reference.
The transformations are useful to calculate how time and distance change for objects moving at significant fractions of the speed of light.

• It connects time and space coordinates so that the speed of light remains the same for everyone, regardless of their motion.
• While the transformation primarily deals with time dilation and length contraction, it's vital in demonstrating that space and time are interwoven into a single continuum known as spacetime.

For instance, if two observers are moving relative to each other, the Lorentz Transformation helps us understand how one observer perceives the time and space measurements taken by the other. This is key when working with high-speed objects that require relativity to explain their behavior accurately.

Speed of Light

The speed of light, denoted as \( c \), is one of the most fundamental constants in physics. Sitting at roughly \( 3 \times 10^8 \text{ meters/second} \), it's pivotal in theories of relativity. Not only is it the speed at which light travels, but it is also the maximum speed at which information and matter can travel in the universe.

• The constancy of the speed of light, regardless of the motion of the light source or observer, is a cornerstone of Einstein’s theory of special relativity.
• This constancy leads to phenomena like time dilation and length contraction, which seem counterintuitive but have been confirmed by numerous experiments.

It is also involved in the famous equation \( E = mc^2 \), showing the equivalence of energy and mass. The speed of light's constant value means that as objects move faster, their behavior diverges from classical physics, requiring relativistic mechanics to describe them accurately.

Velocity Calculation

Calculating the velocity of an object simply means determining how fast and in what direction it is moving. In the context of the exercise with the meter stick, we're interested in finding its velocity relative to an observer using length contraction results.
To find the velocity, we used the rearranged length contraction formula.

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

This gives us insight into how fast the object must be moving to appear contracted to a particular length when viewed by the observer. Substituting the known values into the equation, we get its high-speed movement measured in meters per second.
Understanding the velocity in such problems is essential because it ties together the observable effects predicted by special relativity, like time dilation and length contraction, allowing physicists to make precise predictions about high-speed objects.