Problem 11 - College Physics

Question

A meterstick 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 meterstick to be \(1.00 \mathrm{ft}(1 \mathrm{ft}=0.3048 \mathrm{m})-\) for example, by comparing it with a l-foot ruler that is at rest relative to you, at what speed is the meterstick moving relative to you?

Step-by-Step Solution

Answer

The meterstick is moving at approximately 2.854 x 10^8 m/s relative to you.

Step 1: Understanding the Problem

We need to find the speed of a moving meterstick as it passes by. We know that the measured length is shorter than the actual length due to the special relativity effect known as length contraction.

Step 2: Identify Given Values

The actual length of the meterstick (proper length) is 1 meter. The contracted length we measure is given as 1.00 ft, which is equivalent to 0.3048 meters.

Step 3: Use the Length Contraction Formula

The length contraction formula is given by: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where \(L\) is the contracted length, \(L_0\) is the proper length, \(v\) is the speed of the meterstick, and \(c\) is the speed of light.

Step 4: Substitute Known Values

Substitute the given values into the formula:\[ 0.3048 = 1 \times \sqrt{1 - \frac{v^2}{c^2}} \]

Step 5: Solve for \(v\)

First, simplify the equation:\[ \sqrt{1 - \frac{v^2}{c^2}} = 0.3048 \]Square both sides to remove the square root:\[ 1 - \frac{v^2}{c^2} = 0.0929 \]Rearrange to solve for \(v^2/c^2\):\[ \frac{v^2}{c^2} = 1 - 0.0929 = 0.9071 \]Then, solve for \(v\):\[ v = c \sqrt{0.9071} \]

Step 6: Calculate Velocity

Using \(c = 3 \times 10^8 \text{ m/s}\), \[ v = 3 \times 10^8 \times \sqrt{0.9071} \]Calculate the speed:\[ v \approx 2.854 \times 10^8 \text{ m/s} \]

Step 7: Finalize the Answer

The speed of the meterstick relative to you is approximately \(2.854 \times 10^8 \text{ m/s}\), which is a significant fraction of the speed of light, indicating a relativistic effect.

Key Concepts

Special Relativity

Special relativity, proposed by Albert Einstein, revolutionized our understanding of space and time. It explains how objects behave at high speeds, particularly those approaching the speed of light. At such speeds, time and space are not absolute but relative, meaning they can differ for observers depending on their motion.
Einstein introduced two main postulates in special relativity:

• The laws of physics are the same for all observers, regardless of their relative motion.
• The speed of light in a vacuum is constant and is the same for all observers, regardless of their relative speed.

These ideas lead to several fascinating phenomena, the most notable being time dilation and length contraction.
Length contraction is the key concept in this exercise. It states that objects moving at high speeds will appear shorter along the direction of motion to a stationary observer. This effect only becomes noticeable for objects moving at a significant fraction of the speed of light.

Proper Length

The proper length of an object is a fundamental concept in special relativity. It refers to the length of an object measured in the frame of reference where the object is at rest. In simpler terms, it's how long an object is when it's not moving.
For example, consider a meterstick, like in the exercise. If the meterstick is at rest relative to an observer, its proper length is one meter. However, if the meterstick moves at a very high speed past another observer, this observer would measure the meterstick's length to be less than a meter.
The proper length is crucial in calculating the effects of length contraction. To find the contracted length, you need to know the object's original length when it is stationary. With this information, you can use the length contraction formula to determine the object's apparent length to a moving observer.

Speed of Light

The speed of light in a vacuum, denoted as \( c \), is a universal constant approximately equal to \( 3 \times 10^8 \text{ m/s} \). It is the cosmic speed limit, meaning nothing can exceed this speed. In special relativity, \( c \) plays a critical role, especially in equations like those for length contraction and time dilation.
Because light speed is constant for all observers, regardless of their motion, it becomes a crucial tool for understanding how different observers perceive events in spacetime. In our exercise, we use \( c \) to calculate how much the length of a fast-moving meterstick appears to contract.
It's fascinating to note that as objects approach the speed of light, relativistic effects like length contraction become significant, providing valuable insights into the nature of our universe.

Relativistic Effects

Relativistic effects, including length contraction and time dilation, dramatically influence our perception of objects at high speeds. As per Einstein's theory, these effects manifest significantly when an object's speed approaches the speed of light.
In the exercise, we see length contraction at work. The meterstick, which has a proper length of one meter, appears shorter to an observer standing still as it zooms past them. This occurs due to the length contraction formula derived from special relativity.

• Time dilation: Time passes slower for objects moving at high speeds relative to a stationary observer.
• Mass increase: Objects moving close to the speed of light appear to gain mass.

Understanding these effects helps us comprehend the nature of high-speed travel and shines a light on the fundamental properties of the universe. While such effects are not noticeable in everyday speeds, they are crucial in fields like astrophysics and particle physics.