Question

As a meter stick rushes past me (with velocity v parallel to the stick), I measure its length to be \(80 \mathrm{cm} .\) What is \(v ?\)

Short answer

The velocity \( v \) is \( 0.6c \).

Step-by-step solution

Understand the Problem

We are dealing with length contraction, a relativistic effect that occurs when an object moves at a significant fraction of the speed of light. The length observed is shorter than its proper length when the object is moving.

Write the Length Contraction Formula

The formula for length contraction is given as \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L \) is the observed contracted length, \( L_0 \) is the proper length, \( v \) is the velocity, and \( c \) is the speed of light.

Insert Known Values

Here, \( L = 80 \text{ cm} \) and \( L_0 = 100 \text{ cm} \). Insert these into the length contraction formula: \( 80 = 100 \sqrt{1 - \frac{v^2}{c^2}} \).

Solve for the Velocity v

Start by isolating the square root term: \( \sqrt{1 - \frac{v^2}{c^2}} = \frac{80}{100} = 0.8 \).

Remove the Square Root

Square both sides to eliminate the square root: \( 1 - \frac{v^2}{c^2} = 0.64 \).

Solve for \( \frac{v^2}{c^2} \)

Rearrange the equation: \( \frac{v^2}{c^2} = 1 - 0.64 = 0.36 \).

Calculate v

Finally, solve for \( v \) by taking the square root: \( \frac{v}{c} = \sqrt{0.36} = 0.6 \). Thus, \( v = 0.6c \).

Key concepts

Relativistic Effects

In the realm of physics, especially when dealing with objects moving incredibly fast, the effects of relativity become significant. This is where relativistic effects come into play. These effects were first introduced by Albert Einstein in his Theory of Relativity, fundamentally changing our understanding of space and time.

As objects approach the speed of light, classical mechanics no longer suffices to describe their behavior. Instead, relativistic effects like time dilation, mass increase, and length contraction occur. Length contraction is particularly interesting; it describes how an object in motion appears shorter in the direction of motion to a stationary observer.

In our example, a meter stick moving at high speeds is observed to be only 80 cm long, indicating these relativistic effects. The phenomena only become noticeable when the object's velocity reaches a significant fraction of the speed of light, an occurrence impossible to detect in everyday experiences or slow-moving objects.

• Time Dilation: As speed increases, time appears to slow down for the moving object compared to the observer.
• Length Contraction: Moving objects shrink in the direction of motion when observed from a stationary point.
• Mass Increase: The faster an object moves, the more its mass increases.

Lorentz Transformation

To explain and calculate the changes observed in relativistic effects, we need the Lorentz transformation, a set of equations derived from Einstein's theory. These equations provide a mathematical framework to relate physical quantities between two observers in relative motion.

The Lorentz transformation equations adjust the classical Newtonian mechanics to factor in the finite speed of light, ensuring the constancy of light in all inertial frames. They are essential for describing how lengths contract and times dilate under high-speed circumstances.

• Time Coordinate Adjustment: Modifies time measurements from one observer's frame to another.
• Spatial Coordinate Adjustment: Alters spatial measurements, crucial for analyzing length contraction and other relativistic changes.

In our meter stick scenario, the Lorentz transformations help calculate the stick's contracted length when viewed from a stationary observer's frame of reference. It encapsulates the changes due to velocity, ensuring that all observers, regardless of their relative speed, perceive the speed of light as constant.

Speed of Light

The speed of light, denoted as \(c\), is a fundamental constant in physics, playing a central role in the theory of relativity. Its value is approximately \(299,792,458\) meters per second and is considered the maximum speed at which all energy, matter, and information in the universe can travel.

Einstein's insights showed that the speed of light is identical in all frames of reference, regardless of the motion of the light source or observer. This principle disrupts many intuitive notions about velocity and speed, leading to the aforementioned relativistic effects.

• Universal Constant: The speed of light is the same everywhere and does not change regardless of observer or source movement.
• Upper Speed Limit: Nothing physical can surpass the speed of light, which fundamentally limits speeds in our universe.

In our exercise, the speed of light is crucial because the meter stick's velocity is expressed as a fraction of \(c\). This measurement is necessary to determine when relativistic effects significantly influence observations of speed and distance.

Special Relativity

Special relativity is a theory proposed by Albert Einstein in 1905, aimed at explaining how motion and the laws of physics work when an object travels at speeds close to the speed of light.

This theory leads to several non-intuitive phenomena, like time dilation, length contraction, and the relativity of simultaneity. It implies a remodeling of ordinary concepts of space and time, unifying them into a single "space-time" fabric and ensuring the invariance of physical laws.

• Space-Time: Special relativity merges space and time into one interconnected concept.
• Simultaneity: Events that appear simultaneous in one frame of reference may not appear to occur simultaneously in another.

In our example, special relativity provides the theoretical foundation necessary to understand why a fast-moving meter stick appears shorter than its typical length to a stationary observer. It prompts a shift in classical perspectives, adjusting the interpretation of speed, motion, and distance, which are linked to the constant speed of light.