Problem 23 - University Physics with Modern Physics

Question

An imperial spaceship, moving at high speed relative to the planet Arrakis, fires a rocket toward the planet with a speed of 0.920 c relative to the spaceship. An observer on Arrakis measures that the rocket is approaching with a speed of 0.360 \(\mathrm{c}\) . What is the speed of the spaceship relative to Arrakis? Is the spaceship moving toward or away from Arrakis?

Step-by-Step Solution

Answer

The spaceship is moving away from Arrakis at approximately 0.838c.

Step 1: Understanding the Problem

We are given two speeds: the speed of the rocket relative to the spaceship, 0.920c, and the speed of the rocket as observed on Arrakis, 0.360c. We need to find the speed of the spaceship relative to Arrakis using the relativistic velocity addition formula.

Step 2: Introducing the Relativistic Velocity Addition Formula

The formula for relativistic velocity addition is \( u' = \frac{u+v}{1+\frac{uv}{c^2}} \) where \( u \) is the velocity of the rocket relative to the spaceship, \( v \) is the velocity of the spaceship relative to Arrakis, and \( u' \) is the velocity of the rocket relative to Arrakis.

Step 3: Solving for the Spaceship's Speed

Plug in the known values: \( u = 0.920c \) (rocket's speed relative to the spaceship) and \( u' = 0.360c \) (speed observed by Arrakis). We solve for \( v \):\[ 0.360 = \frac{0.920 + v}{1 + \frac{0.920v}{c^2}} \]

Step 4: Clearing the Denominator

Multiply both sides by the denominator to clear it:\[ 0.360 (1 + \frac{0.920v}{c^2}) = 0.920 + v \]

Step 5: Expanding and Rearranging the Equation

Expand the left side:\[ 0.360 + 0.3312v = 0.920 + v \]Rearrange terms to isolate \( v \):\[ v - 0.3312v = 0.920 - 0.360 \]

Step 6: Simplifying the Equation

Simplify the terms:\[ 0.6688v = 0.560 \]

Step 7: Solving for v

Solve for \( v \):\[ v = \frac{0.560}{0.6688} \approx 0.8375 \]Thus, the speed of the spaceship relative to Arrakis is approximately 0.838c.

Step 8: Determining the Spaceship's Direction

Since the observed speed of the rocket is less than the speed relative to the spaceship, the spaceship is moving away from Arrakis.

Key Concepts

Velocity of the Spaceship

When dealing with the concept of relativistic speeds, it is crucial to understand how the velocity of an object, like a spaceship, is perceived differently depending on the reference frame. In this example, the spaceship is traveling relative to the planet Arrakis. To determine this velocity, we must use the relativistic velocity addition formula, which helps calculate speeds in a relativistic context where speeds are significant fractions of the speed of light ("c").

The velocity addition formula is expressed as:

• \( u' = \frac{u+v}{1+\frac{uv}{c^2}} \)

Here:

• \( u \) is the speed of the moving object (the rocket) relative to the spaceship.
• \( v \) is the speed of the spaceship relative to Arrakis.
• \( u' \) is the velocity of the rocket as observed from Arrakis.

By substituting known values into this formula, we can solve for the unknown, which is the spaceship’s velocity relative to Arrakis. In our case, the solution gives us \( v \approx 0.838c \).

This means the spaceship travels at 0.838 times the speed of light when viewed from the planet.

Speed of Light

The speed of light, denoted by "c," is approximately 299,792,458 meters per second in a vacuum. This value is significant in the realm of physics, especially when talking about relativistic speeds. The importance of the speed of light emerges in Einstein's theory of relativity, which fundamentally alters how we understand motion.

When calculating velocities in a relativistic context, such as how fast the spaceship is moving from Arrakis’ point of view, the speed of light becomes the ultimate speed limit for any object with mass. Speeds nearing that of light require us to use Einstein’s relativistic velocity addition, rather than simple arithmetic addition, because measurements become non-linear as they approach "c." The intricacies of Einstein's theory imply:

• Velocities add differently; speeds do not merely sum up.
• As speeds increase towards "c," their addition results in speeds asymptotically approaching, but never actually reaching "c."

Observer Reference Frames

In physics, reference frames significantly influence the way motion, speed, and time are perceived. A reference frame is essentially a perspective from which measurements and observations are made. In our problem, there are two main reference frames: the spaceship and the planet Arrakis.

The spaceship's reference frame is where the rocket is launched, and it sees the rocket traveling at 0.920c. Meanwhile, Arrakis represents a stationary reference frame from which the rocket's speed is observed to be 0.360c.

Using these two reference frames, we rely on the relativistic velocity addition formula to understand how objects moving at significant fractions of the speed of light are perceived differently from various viewpoints.

Key insights around observer reference frames include:

• Each observer measures velocities uniquely based on their own frame.
• In relativistic conditions, time dilation and length contraction affect perceived velocities.
• Switching reference frames alters how we calculate speeds and understand motion dynamics.

Understanding these differences is crucial in scenarios like space travel, where high velocities are common and traditional physics does not suffice.