Problem 30 - Fundamentals of Physics

Question

Stellar system \(Q_{1}\) moves away from us at a speed of \(0.800 c\). Stellar system \(Q_{2}\), which lies in the same direction in space but is closer to us, moves away from us at speed \(0.400 c\). What multiple of c gives the speed of \(Q_{2}\) as measured by an observer in the reference frame of \(Q_{1}\) ?

Step-by-Step Solution

Answer

The speed of \(Q_{2}\) relative to \(Q_{1}\) is approximately \(-0.588c\).

Step 1: Identify the Problem

We need to find the relative speed of stellar system \(Q_{2}\) as observed from \(Q_{1}\), given their speeds relative to us (the Earth). The speeds are given as fractions of the speed of light, \(c\).

Step 2: Use the Relativistic Velocity Addition Formula

The formula to use in this scenario is the relativistic velocity addition formula: \[ u' = \frac{u - v}{1 - \frac{uv}{c^2}} \]where \(u\) is the speed of \(Q_{2}\) as seen by us (\(0.400c\)), \(v\) is the speed of \(Q_{1}\) relative to us (\(0.800c\)), and \(u'\) is the speed of \(Q_{2}\) as seen from \(Q_{1}\).

Step 3: Plug Values into the Formula

Substitute the values into the formula:\[ u' = \frac{0.400c - 0.800c}{1 - \frac{(0.400c)(0.800c)}{c^2}} \]=\[\frac{-0.400c}{1 - 0.32}\].

Step 4: Simplify the Expression

Simplify the denominator:\[ u' = \frac{-0.400c}{0.68} \].

Step 5: Calculate the Result

Perform the division to find the speed of \(Q_{2}\) as measured by \(Q_{1}\):\[ u' \approx -0.588c \]. This negative sign indicates that from \(Q_{1}\)'s perspective, \(Q_{2}\) is moving in the opposite direction.

Key Concepts

Special Relativity

Special Relativity is a theory developed by Albert Einstein, which fundamentally changed the way we understand space and time. At its core, it deals with how observers moving at different velocities perceive phenomena in the universe. The key postulate is that the laws of physics are the same for all observers, regardless of their speed of motion, and that the speed of light in vacuum is constant for all observers, regardless of the motion of the light source or observer. This theory leads to some intriguing effects, such as time dilation and length contraction, which become significant at velocities close to the speed of light.

• Time dilation: Moving clocks tick slower than stationary ones, as observed from the stationary frame.
• Length contraction: Objects moving at a high velocity appear shorter along the direction of motion from the perspective of a stationary observer.

In the context of the stellar problem given, Special Relativity gives us the framework to understand how the velocities of the stellar systems are perceived differently because of their significant fractions of the speed of light.

Reference Frames

A reference frame is a perspective from which an observer measures and experiences the universe around them. In physics, choosing the right reference frame is crucial for solving problems accurately. It allows us to determine how objects move and the speeds at which they travel. An inertial reference frame is one that is either at rest or moves at a constant velocity.

• Each inertial reference frame is considered equally valid by the laws of physics.
• Observers in different reference frames might perceive space and time differently because of their relative motion.

In this exercise, the Earth is one reference frame, while each stellar system (\(Q_1\) and \(Q_2\)) acts as another. The relativistic velocity addition formula used here accounts for the transformations needed to understand how an observer in \(Q_1\) measures the speed of \(Q_2\). This process highlights the importance of understanding and using reference frames appropriately when dealing with relativistic speeds.

Speed of Light

The speed of light, denoted as \(c\), is a critical constant in physics, valued at approximately 299,792,458 meters per second. It represents the maximum speed at which information or matter can travel in the universe. The constancy and limit imposed by the speed of light are fundamental to the theory of Special Relativity.

• The speed of light is the same for all observers, regardless of their motion relative to the light source. This is a cornerstone of Einstein’s theory.
• In relativistic physics, speeds are often expressed as fractions of \(c\), such as 0.400c or 0.800c, indicating significant portions of light speed.

In problems involving relativistic velocities, such as our stellar systems moving away at speeds of 0.800c and 0.400c, it is essential to apply formulas like the relativistic velocity addition. These ensure that the combined speeds never exceed the speed of light, maintaining the universal speed limit set by \(c\). In our exercise, using the formula reveals how speeds combine differently than our everyday experiences would suggest, due to these relativistic effects.