The starships of the Solar Federation are marked with the...

The starships of the Solar Federation are marked with the symbol of the federation, a circle, while starships of the Denebian Empire are marked with the empire's symbol, an ellipse whose major axis is 1.40 times longer than its minor axis ($a$ = 1.40$b$ in $\textbf{Fig. P37.45}$). How fast, relative to an observer, does an empire ship have to travel for its marking to be confused with the marking of a federation ship?

The starships of the Solar Federation are marked with the symbol of the federation, a circle, while starships of the Denebian Empire are marked with the empire's symbol, an ellipse whose major axis is
1.40 times longer than its minor axis ($a$ = 1.40$b$ in $\textbf{Fig. P37.45}$). How fast, relative to an observer, does an empire ship have to travel for its marking to be confused with the marking of a federation ship?

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Key Concepts

Lorentz Factor

The Lorentz factor, defined as ? = 1/?(1 - v²/c²), quantifies how much time, length, and relativistic mass change due to high-speed motion. It plays a key role in determining the amount of length contraction experienced by an object moving relative to an observer.

Relativistic Reference Frames

In the context of special relativity, the measurements of space and time depend on the observer's inertial frame. An object in one frame may appear contracted or dilated when viewed from another frame moving at a significant fraction of the speed of light, highlighting the relative nature of measurements in different frames.

Length Contraction

In special relativity, objects moving at high speeds appear shortened along the direction of motion relative to their rest length. This phenomenon, known as length contraction, means that dimensions parallel to the velocity are reduced by a factor of ?(1 - v²/c²), while dimensions perpendicular to the motion remain unchanged.

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