Problem 33 - Fundamentals of Physics

Question

An armada of spaceships that is 1.00 ly long (as measured in its rest frame) moves with speed \(0.800 \mathrm{c}\) relative to a ground station in frame \(S\). A messenger travels from the rear of the armada to the front with a speed of \(0.950 \mathrm{c}\) relative to \(S .\) How long docs 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

(a) 0.197 years, (b) 1.533 years, (c) 0.632 years.

Step 1: Understand the Given Information

We have an armada of spaceships that is 1 light-year (ly) long in its rest frame moving at a speed of 0.800c relative to a frame \(S\). A messenger travels from the rear to the front of the armada with a speed of 0.950c relative to frame \(S\). We are to find the travel time of the messenger in three different frames.

Step 2: Calculate the Length of Armada in Ground Frame

The armada's contracted length in frame \(S\) can be found using the Lorentz contraction formula: \(L = L_0 \sqrt{1 - v^2/c^2}\), where \(L_0 = 1.00\) ly is the proper length and \(v = 0.800c\) is the speed of the armada relative to \(S\). Substitute the values: \[ L = 1.00 \sqrt{1 - (0.800)^2} = 1.00 \sqrt{1 - 0.640} = 1.00 \sqrt{0.360} = 0.600 \text{ ly} \] This is the length of the armada as measured by an observer in frame \(S\).

Step 3: Calculate Time in Ground Frame

In frame \(S\), the messenger travels 0.600 ly at a speed of 0.950c. The time taken is given by \(t = \frac{L}{v}\).Substitute the values: \[ t_S = \frac{0.600 \text{ ly}}{0.950c} \]Let 1 ly/c = 1 year, giving: \[ t_S = \frac{0.600}{0.950} = 0.632 \text{ years} \]This is the time as measured in the ground frame \(S\).

Step 4: Calculate Time in Messenger's Rest Frame

Use the time dilation formula: \(t_m = \frac{t_S}{\gamma}\), where \(\gamma = \frac{1}{\sqrt{1 - v_m^2/c^2}}\) and \(v_m = 0.950c\) is the messenger's speed relative to the ground.First, calculate \(\gamma\): \[ \gamma = \frac{1}{\sqrt{1 - (0.950)^2}} = \frac{1}{\sqrt{0.0975}} \approx 3.202 \]Therefore, the time in the messenger's rest frame is:\[ t_m = \frac{0.632}{3.202} = 0.197 \text{ years} \]

Step 5: Calculate Time in Armada's Rest Frame

Since the armada is at rest in its own frame, the length is 1.00 ly, and the messenger is moving with speed \(v_{ma}\) relative to the armada, where \(v_{ma}\) is found using the relativistic velocity addition formula: \[ v_{ma} = \frac{v_m - v}{1 - \frac{v_mv}{c^2}} = \frac{0.950c - 0.800c}{1 - (0.950)(0.800)}c \approx 0.652c \]Now, use \(t_a = \frac{L_0}{v_{ma}} \):\[ t_a = \frac{1.00 \text{ ly}}{0.652c} \approx 1.533 \text{ years} \]

Key Concepts

Lorentz contraction

Lorentz contraction is an intriguing concept that arises from Einstein's theory of special relativity. It refers to the phenomenon where the length of an object moving relative to an observer is measured to be shorter than its proper length (the length measured in the object's rest frame). This contraction occurs only in the direction of the relative motion.

To illustrate with our example, the armada of spaceships, when stationary in their own rest frame, measures 1.00 light-year (ly) long. However, when moving at 0.800 times the speed of light (c) relative to a ground observer, the length appears shorter due to Lorentz contraction. The formula to calculate this contracted length is:\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where:

• \( L \) is the contracted length as measured by the observer.
• \( L_0 \) is the proper length of the object.
• \( v \) is the velocity of the object relative to the observer.
• \( c \) is the speed of light.

For the armada, the contracted length calculated is 0.600 ly, illustrating how fast-moving objects appear compressed in the direction of motion.

Time dilation

Another fundamental concept in relativity is time dilation. This occurs because the passage of time is relative and can vary depending on the relative velocity between observers. In the frame of the object moving at high velocities, time appears to "dilate" or slow down when viewed from a stationary frame.

Let's consider the messenger's journey from the rear to the front of the armada. When observed from the ground station (frame \( S \)), the time taken is 0.632 years. However, for the messenger, who is moving and experiences time differently, the journey duration is less.

This is calculated using the time dilation formula:\[ t_m = \frac{t_S}{\gamma} \]where \( \gamma \) (Lorentz factor) is given by:\[ \gamma = \frac{1}{\sqrt{1 - \frac{v_m^2}{c^2}}} \]

• \( t_m \) is the time in the moving frame (messenger's frame).
• \( t_S \) is the time in the stationary frame (ground frame).
• \( v_m \) is the velocity of the messenger.

For the messenger, this results in a time of about 0.197 years. This reflects how the faster you travel, the more time seems to "slow down" from the perspective of a stationary observer.

Relativistic velocity addition

In relativity, the way velocities are added is not as straightforward as in classical physics, particularly when dealing with speeds near that of light. This is where the concept of relativistic velocity addition comes into play, ensuring that computed velocities do not exceed the speed of light.

To compute the speed of the messenger as seen in the armada's rest frame, we use the relativistic velocity addition formula:\[ v_{ma} = \frac{v_m - v}{1 - \frac{v_mv}{c^2}} \]where:

• \( v_{ma} \) is the messenger's velocity relative to the armada.
• \( v_m \) is the messenger's velocity relative to the ground station.
• \( v \) is the armada's velocity relative to the ground station.

In the example, this results in a messenger speed of approximately 0.652c relative to the armada. This calculated speed makes sense because it ensures that even the summed velocities in the relativistic framework adhere to the constant speed of light limiting principle.

Ultimately, relativistic velocity addition provides a consistent way to handle the complexities of high-speed motion, revealing the profound effects of relativity on how we perceive motion and time.