Question

An observer in frame \(S'\) is moving to the right (+\(x\)-direction) at speed \(u\) = 0.600c away from a stationary observer in frame S. The observer in \(S'\) measures the speed \(v'\) of a particle moving to the right away from her. What speed \(v'\) does the observer in S measure for the particle if (a) \(v'\) = 0.400c; (b) \(v'\) = 0.900c; (c) \(v'\) = 0.990c?

Short answer

(a) 0.806c, (b) 0.974c, (c) 0.997c

Step-by-step solution

Understanding the Problem

We need to find the speed \( v \) of a particle as observed from the frame \( S \), given the speed \( v' \) from frame \( S' \), where frame \( S' \) moves at speed \( u = 0.600c \). We'll use the relativistic velocity addition formula.

Utilizing Relativistic Velocity Addition

The formula for relativistic velocity addition is given by: \[ v = \frac{v' + u}{1 + \frac{v' u}{c^2}} \] where \( v \) is the velocity of the particle as seen from frame \( S \), \( v' \) is the velocity of the particle as seen from frame \( S' \), and \( u \) is the speed of \( S' \) relative to \( S \).

Case (a): Calculate Velocity for \( v' = 0.400c \)

Substitute \( v' = 0.400c \) and \( u = 0.600c \) into the formula: \[ v = \frac{0.400c + 0.600c}{1 + \frac{0.400c \times 0.600c}{c^2}} = \frac{1.000c}{1 + 0.240} = \frac{1.000c}{1.240} = 0.806c \].

Case (b): Calculate Velocity for \( v' = 0.900c \)

Substitute \( v' = 0.900c \) and \( u = 0.600c \): \[ v = \frac{0.900c + 0.600c}{1 + \frac{0.900c \times 0.600c}{c^2}} = \frac{1.500c}{1 + 0.540} = \frac{1.500c}{1.540} \approx 0.974c \].

Case (c): Calculate Velocity for \( v' = 0.990c \)

Substitute \( v' = 0.990c \) and \( u = 0.600c \): \[ v = \frac{0.990c + 0.600c}{1 + \frac{0.990c \times 0.600c}{c^2}} = \frac{1.590c}{1 + 0.594} = \frac{1.590c}{1.594} \approx 0.997c \].

Final Results

For \( v' = 0.400c \), \( v = 0.806c \); for \( v' = 0.900c \), \( v \approx 0.974c \); for \( v' = 0.990c \), \( v \approx 0.997c \).

Key concepts

Lorentz Transformation

The Lorentz transformation is a fundamental cornerstone of special relativity. It mathematically describes how observations made in different reference frames, particularly those moving relative to each other at high velocities, relate. This transformation includes formulas for how time and space coordinates change between frames. For velocities, it specifically helps us understand how speed is perceived differently by observers in different frames. In the context of our exercise, the Lorentz transformation provides the relativistic velocity addition formula. This formula allows one to calculate the velocity of a particle in one frame (e.g., a stationary frame) when it is known in another frame (e.g., a moving frame). It's crucial when dealing with speeds close to that of light because classical addition of velocities doesn't apply. Key aspects of Lorentz transformations:

• Solves discrepancies between observers in different frames.
• Ensures the speed of light is constant in all frames, as per Einstein's postulate.
• Ensures correct transformations of velocities, times, and positions between frames.

Understanding and applying the Lorentz transformation is vital to analyzing scenarios involving high-speed particles like those in the given exercise.

Special Relativity

Special Relativity, introduced by Albert Einstein in 1905, revolutionizes our understanding of physics, especially about time and space for observers moving at constant speeds relative to each other. This theory postulates that the laws of physics are the same for all non-accelerating observers, and notably, that the speed of light in a vacuum is the same regardless of the speed of the observer or source. This has profound implications, especially when dealing with high-velocity particles. In the exercise, special relativity is key because it allows us to use the relativistic velocity addition formula, which is a result of its postulates. Without special relativity, our usual intuition about adding velocities would fail at speeds approaching that of light. Some principles of Special Relativity to remember:

• Time dilation: Moving clocks run slower compared to stationary ones.
• Length contraction: Objects appear shorter in the direction of motion from the perspective of a moving observer.
• Simultaneity: Two events that appear concurrent in one frame may not be so in another.

Special relativity challenges our everyday experiences and broadens understanding, especially in the domain of astrophysics.

Reference Frames

Reference frames are like the stage where the physics of motion and interactions occur, allowing observers to measure and perceive physical quantities like position, velocity, and time. Each observer has their own reference frame, which can be either stationary or moving. In our exercise, two reference frames are involved: frame S (stationary) and frame S' (moving at 0.600c). Each frame provides a unique view of the universe, and transitioning from one frame to another involves transformations that consider the relative motion between the frames. Key insights about reference frames:

• Absolute motion doesn't exist; motion is always relative to a reference frame.
• In Newtonian physics, reference frames are straightforward, but in relativity, transformations between frames are intricate.
• Changes between reference frames in relativity can result in counterintuitive observations such as time dilation and length contraction.

Understanding reference frames helps unravel diverse perspectives in physics, allowing for accurate measurements and insights when velocities are extreme, as demonstrated in the velocity addition problem."}]}]}