Problem 33 - Fundamentals of Physics

Question

(a) An armada of spaceships that is \(1.00\) ly long (as measured in its rest frame) moves with speed \(0.800 c\) relative to a ground station in frame \(S .\) A messenger travels from the rear of the armada to the front with a speed of \(0.950 c\) relative to \(S .\) How long does the trip take as measured (a) in the rest frame of the messenger, (b) in the rest frame of the armada, and (c) by an observer in the ground frame \(S ?\)

Step-by-Step Solution

Answer

In frame S, the length of armada is 0.600 ly.

Step 1: Identify the physical scenario

List given quantities and unknowns.

Step 2: Determine relevant principles

Identify applicable physics laws.

Step 3: Set up and solve

Write equations and solve.

Step 4: State the answer

The answer is: In frame S, the length of armada is 0.600 ly.

Key Concepts

Length Contraction

Length contraction is a phenomenon that occurs when an object is moving very fast, close to the speed of light, relative to an observer. It's an essential concept in the theory of relativity that helps us understand how objects appear shorter along the direction of their motion from the perspective of an observer.

To calculate the contracted length, we use the formula: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where:

• \( L_0 \) is the proper length (length of the object in its rest frame).
• \( v \) is the velocity of the moving object relative to the observer.
• \( c \) is the speed of light.

This formula tells us that the faster an object travels relative to the speed of light, the more significant the contraction becomes. In our example, the armada, which is 1.00 light-year long in its rest frame, measures only 0.600 light-years from the ground observer's frame due to its high speed of 0.800 times the speed of light.

Proper Length

Proper length is the length of an object measured in the object's own rest frame. This is the length when the object is not moving relative to the observer. It's important because it represents the true, uncontracted length of the object.

For example, if you're measuring the length of a spaceship as you travel with it, that measurement is the proper length. In the armada's case, its proper length is 1.00 light-year because that's how long it is when measured from its own rest frame, where it is stationary.

Proper length remains constant, unlike the contracted length, which varies depending on the relative speed of an observer. This is why it's crucial to identify whether we're talking about proper length or contracted length in physics problems.

Speed of Light

The speed of light, denoted as \( c \), is a fundamental constant in physics, approximately \( 3 \times 10^8 \text{ m/s} \) or about 186,282 miles per second. It's the maximum speed at which information or matter can be transmitted in the universe.

In the realm of relativity, the speed of light is central. It acts as the universal "speed limit," influencing how time and space behave at high velocities. For example, in our exercise, the armada travels at 0.800 times the speed of light. Even at just a fraction of the speed of light, relativity's effects become noticeable, leading to phenomena like time dilation and length contraction.

Relativity posits that the speed of light is the same for all observers, no matter their relative motion. This principle leads to many non-intuitive outcomes but is crucial for understanding physics at cosmological scales.

Reference Frames

In physics, a reference frame is a viewpoint for observing and measuring physical phenomena, like motion and time. It's a system that helps define the position and speed of objects in the universe.

For example, in the exercise, we deal with several reference frames:

• The rest frame of the messenger, who measures time for his travel without any relativity effects.
• The rest frame of the armada, where proper length is used, and the armada is stationary.
• The ground frame \( S \), where the armada moves at 0.800 times the speed of light relative to the observer.

Each reference frame provides a unique perspective on the same event, leading to different measurements of time and space. Understanding these perspectives is crucial when tackling relativity problems because it explains why observers in different frames might disagree on measurements like time, distance, and speed.