Problem 27 - Fundamentals of Physics

Question

A particle moves along the \(x\) " axis of frame \(S^{\prime}\) with velocity \(0.40 \mathrm{c}\). Frame \(S^{\prime}\) moves with velocity \(0.60 \mathrm{c}\) with respect to frame \(S .\) What is the velocity of the particle with respect to frame \(S ?\)

Step-by-Step Solution

Answer

The velocity of the particle with respect to frame \( S \) is approximately \( 0.806c \).

Step 1: Identify Given Information

We have two velocities given: the velocity of the particle in frame \( S' \), which is \( v_{p'} = 0.40c \), and the velocity of frame \( S' \) with respect to \( S \), which is \( v_{S'} = 0.60c \). Here, \( c \) is the speed of light.

Step 2: Use the Velocity Addition Formula

We need to find the velocity of the particle relative to frame \( S \) using the relativistic velocity addition formula: \( v = \frac{v_{p'} + v_{S'}}{1 + \frac{v_{p'} v_{S'}}{c^2}} \). This formula accounts for the effects of special relativity when adding velocities.

Step 3: Plug in the Values

Substitute \( v_{p'} = 0.40c \) and \( v_{S'} = 0.60c \) into the formula: \[ v = \frac{0.40c + 0.60c}{1 + \frac{0.40c \times 0.60c}{c^2}}.\] Simplify to get \[ v = \frac{1.00c}{1 + 0.24}.\]

Step 4: Simplify the Equation

Calculate the denominator: \( 1 + 0.24 = 1.24 \). Then, divide the numerator by the denominator: \[ v = \frac{1.00c}{1.24}.\]

Step 5: Compute the Final Velocity

Perform the division to find \( v \): \[ v \approx 0.806c.\] Thus, the velocity of the particle with respect to frame \( S \) is approximately \( 0.806c \).

Key Concepts

special relativity

The theory of special relativity, proposed by Albert Einstein in 1905, revolutionized our understanding of physics by introducing ideas of space and time being interconnected. Before special relativity, it was assumed that the laws of physics were identical for all frames of reference moving at constant speeds relative to each other. However, special relativity extended this principle to include scenarios involving near-light speeds.

This groundbreaking theory introduced two major postulates:

• The laws of physics are invariant in all inertial frames of reference (i.e., non-accelerating frames).
• The speed of light in a vacuum is the same for all observers, regardless of the motion of light source or observer.

These postulates laid the foundation for understanding phenomena such as time dilation and length contraction, which occur when objects move at significant fractions of the speed of light, greatly affecting our observations in different reference frames.

velocity in different reference frames

In the realm of special relativity, velocities are not simply additive as they are in classical physics. Instead, we must consider the relativistic velocity addition formula. This formula helps us calculate the velocity of an object as observed from a different reference frame. This consideration becomes particularly important when dealing with objects moving at speeds close to that of light.

The relativistic velocity addition formula is given by:\[v = \frac{v_{1} + v_{2}}{1 + \frac{v_{1} v_{2}}{c^2}}\]where:

• \(v\) is the resultant velocity observed in the new reference frame.
• \(v_{1} \) and \(v_{2} \) are the velocities that are being added.
• \(c\) is the constant speed of light.

The formula accounts for relativistic effects, ensuring no resultant velocity exceeds the speed of light, which is consistent with the principles of special relativity.

speed of light

The speed of light, denoted as \(c\), is fundamental to special relativity and is one of the most important constants in physics. Light travels at approximately \(3 \times 10^8\) meters per second (or about 299,792 kilometers per second) in a vacuum. This speed is exceedingly fast compared to everyday speeds and is considered the ultimate speed limit in the universe.

Several crucial aspects of the speed of light make it unique:

• It remains constant across all inertial frames, serving as a universal benchmark for comparing velocities.
• It is independent of the motion of the source or observer, an idea that radically shifts the classical understanding of speed.
• It sets a limitation where no information or matter can travel faster than light, ensuring causality is preserved in the universe.

These attributes of the speed of light significantly impact our understanding of time, space, and energy, forming a cornerstone of modern physics.