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Special relativity is a theory developed by Albert Einstein that revolutionized our understanding of space and time. It emphasizes how different observers perceive the same event when they are moving at high speeds relative to one another. One of the key ideas in special relativity is that the speed of light is the same for all observers, regardless of their relative motion. This contrasts with classical mechanics and leads to phenomena like time dilation and length contraction. In the context of our problem, special relativity affects how velocities are added. If we used the classical approach, we might expect to simply add velocities. However, at high speeds, particularly those close to the speed of light, special relativity requires us to use a more complex formula to calculate the relative velocity between moving objects.

Calculating velocity in the framework of special relativity involves more than simple addition or subtraction. We rely on the relativistic velocity addition formula. This formula accounts for the fact that as objects approach the speed of light, their velocities don't just add up in a straightforward way.
In the exercise, we use the formula: \[ u = \frac{u' + v}{1 + \frac{u'v}{c^2}} \]where:

• \(u'\) is the velocity of the rocket relative to the spaceship,
• \(v\) is the velocity of the spaceship relative to Arrakis,
• \(u\) is the velocity of the rocket relative to Arrakis.

Simply putting the speeds into this equation gives us the tools to determine the spaceship's speed relative to Arrakis. The complexity comes from the need to consider how velocities interact at these high speeds.

The concept of reference frames is crucial in physics, especially in special relativity. A reference frame is essentially a perspective from which an observer or a set of observers measure and interpret physical events. Each reference frame comes with its own view of space and time.
In our problem, we have two main reference frames: one attached to the spaceship and the other to the planet Arrakis. The observer on Arrakis sees the rocket's speed as 0.360\(c\), while the spaceship sees it as 0.920\(c\). Understanding how these differing perspectives relate is the essence of relativistic calculations. In practice, this means that the motions observed in one frame need adjustments using the relativistic formulas to be interpreted correctly in another. This principle ensures consistency among different observers, even when they are in relative motion to each other at significant fractions of the speed of light.