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In physics, frequency is a fundamental concept used to describe how often an event occurs over a specific period of time. In the context of the given exercise, frequency is defined as the number of cycles or, in this case, the number of times the lighthouse beam completes one full circle in one second.
- The frequency observed from Earth is referred to as the actual frequency.- The frequency observed from the spaceship, which is moving away from Earth, is the apparent frequency.
The relationship between time and frequency is inverse: as one increases, the other decreases. The actual frequency of the lighthouse beam is calculated by the formula:\[ f_\text{actual} = \frac{1}{7.5 \, \text{s}} = \frac{1}{7.5} \, \text{Hz} \]The apparent frequency, as seen from the spaceship, is given by:\[ f_\text{apparent} = \frac{1}{15 \, \text{s}} = \frac{1}{15} \, \text{Hz} \]The difference between these frequencies is due to the relativistic Doppler effect, which occurs because the spaceship is moving away from the source of the light.

The speed of light is a constant in vacuum and is crucial for understanding various principles in physics, including the Doppler effect. In a vacuum, the speed of light, denoted by \( c \), is approximately \( 3 \times 10^8 \, \text{m/s} \). It is one of the universal constants in physics and plays a vital role in calculations involving relativistic physics.
When calculating changes in observed frequency due to relative motion, it is important to use the speed of light accurately as it impacts the observed versus actual measurements of the frequency.
In the context of the relativistic Doppler effect, the apparent frequency change accounts for the finite speed of light when interpreting how motion affects eyewitness observations of time and space between communicating objects such as spaceships and lighthouses. The speed of light is vital in these calculations to ensure results are consistent with the laws of physics.

Calculating the velocity of an object, like the spaceship in this exercise, often involves using known values and solving equations such as those provided by the Doppler effect. Here, we must find the speed of the spaceship, taking into account the speeds of light and observable frequency changes.
The Doppler effect formula for light can be rearranged to solve for the spaceship's velocity \( v \): \[ f_\text{apparent} = f_\text{actual} \times \sqrt{\frac{1 - v/c}{1 + v/c}} \]Rearrange this equation to isolate the velocity term:\[ \sqrt{\frac{1 - v/c}{1 + v/c}} = \frac{f_\text{apparent}}{f_\text{actual}} \]By simplifying and solving, we find that \[ \frac{1 - v/c}{1 + v/c} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \],which further simplifies to:\[ v = \frac{3}{5} c \]This gives us a velocity, which indicates how fast the spaceship is moving away from the Earth. Inserting the constant speed of light \( c \), we calculate: \[ v = 0.6 \times 3 \times 10^8 \, \text{m/s} = 1.8 \times 10^8 \, \text{m/s} \]Understanding velocity calculation in this context is critical for determining the spaceship's speed relative to Earth.