Problem 88 - Fundamentals of Physics

Question

A Foron cruiser moving directly toward a Reptulian scout ship fires a decoy toward the scout ship. Relative to the scout ship, the speed of the decoy is \(0.980 \mathrm{c}\) and the speed of the Foron cruiser is \(0.900 \mathrm{c}\). What is the speed of the decoy relative to the cruiser?

Step-by-Step Solution

Answer

The speed of the decoy relative to the cruiser is approximately \(0.678c\).

Step 1: Identify the Problem Type

The exercise involves relative velocities in the context of special relativity. Here, we want to find the relative speed of the decoy with respect to the Foron cruiser using initial given velocities.

Step 2: Set Up Formula for Relativistic Velocity Addition

To find the speed of the decoy relative to the cruiser, use the relativistic velocity addition formula: \( v' = \frac{v - u}{1 - \frac{uv}{c^2}} \), where \( v \) is the velocity of the decoy relative to the scout, \( u \) is the velocity of the Foron cruiser relative to the scout, and \( c \) is the speed of light.

Step 3: Substitute Known Values

Substitute the given values into the relativistic velocity addition formula: \( v = 0.980c \), \( u = 0.900c \), and \( c = c \): \[ v' = \frac{0.980c - 0.900c}{1 - \frac{(0.980c)(0.900c)}{c^2}} \]

Step 4: Simplify the Equation

Simplify the equation by performing the arithmetic calculations: \[ v' = \frac{0.080c}{1 - 0.882} \]

Step 5: Calculate the Final Result

Calculate the value: \[ v' = \frac{0.080c}{0.118} \approx 0.678c \] This gives the speed of the decoy relative to the cruiser.

Key Concepts

Relativistic Velocity Addition

In the realm of special relativity, the rules for adding velocities are quite different from what you may have learned in classical physics. When two objects are moving at speeds close to the speed of light (\(c\)), we must use the relativistic velocity addition formula to determine their relative velocity. This formula is crucial to handle scenarios where high velocities make it impossible to simply add or subtract speeds in the straightforward manner used in classical mechanics.
To understand the relativistic velocity addition, consider the formula:

• \[ v' = \frac{v - u}{1 - \frac{uv}{c^2}} \]

Here, \(v\) is the velocity of one object (the decoy, in our example) relative to a third reference point (the scout ship), \(u\) is the velocity of another object (the Foron cruiser) relative to the same reference point, and \(v'\) is the velocity of the first object (the decoy) relative to the second (the Foron cruiser).
What makes this formula special is its ability to keep the resulting velocity below the speed of light, respecting nature's ultimate speed limit. This ensures that even if two speeds are close to \(c\), their combined effect will never exceed it.

Relative Velocity

Relative velocity is a concept dealt with not only in special relativity but also in everyday scenarios. It essentially measures how fast one object is moving concerning another. While the basic concept remains the same, in the context of special relativity, calculations get much more interesting.
In our case, we want to determine how fast the decoy is moving relative to the Foron cruiser. Unlike classical views, which would simply subtract one speed from the other, relativistic scenarios require careful consideration of time and space distortions at high speeds.
When dealing with velocities approaching the speed of light, you must account for the effects discussed in Einstein's theory of relativity. It affects how time and distances are perceived, altering how we observe motion and speed. This is why we turn to the relativistic velocity addition formula, allowing us to factor in these immense speeds and providing the correct relative velocity value.

Speed of Light

The speed of light (\(c\)) is not just a number; it's a fundamental constant of nature. Measured at approximately 299,792,458 meters per second, it serves as the default speed limit for the universe. Nothing can travel faster than light, setting a definitive ceiling for how fast anything can go.
In special relativity, the speed of light remains a constant regardless of the observer's motion. Whether you're in a moving vehicle or stationary, any light you observe will always travel at the speed of light. This constancy is what makes Einstein's theory both revolutionary and indispensable in understanding high-speed phenomena.
All the principles of special relativity, including the relativistic addition of velocities, are built on the constancy of light's speed. Whether calculating relative velocities or understanding time dilation and length contraction, the speed of light remains central to any discussions around special relativity.