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Length contraction is a fascinating concept from Einstein's theory of special relativity. It proposes that an object in motion appears shorter along the direction of motion when observed from a stationary frame. This means as a spaceship travels close to the speed of light, it looks shorter to an observer who isn't moving with it. The formula that describes this is:

• \( L = L_0 \sqrt{1 - \beta^2} \)

Here, \( L \) is the observed length, \( L_0 \) is the rest length, and \( \beta = \frac{v}{c} \), where \( v \) is the velocity of the object and \( c \) is the speed of light.
In the given exercise, the spaceship's length appears to be half of its rest length. We can make an equation out of this:

• \( \frac{1}{2}L_0 = L_0 \sqrt{1 - \beta^2} \)

By solving this equation, you can find the spaceship's speed as a fraction of the speed of light.

Time dilation is another fascinating result of the theory of special relativity. It describes how time can appear to "slow down" for an object moving close to the speed of light, relative to an observer at rest. This means that if you were traveling on a fast-moving spaceship, time would pass slower for you compared to someone stationary, observing from outside.

• The formula is:\[ t' = \frac{t}{\sqrt{1 - \beta^2}} \]
• \( t' \) is the time measured in the moving frame, \( t \) is the time in the rest frame, and \( \beta = \frac{v}{c} \).

In the exercise, knowing \( \beta \) helps calculate the time dilation factor. If the spaceship’s \(\beta\) is approximately 0.866, the clocks on board run slower by a factor of 2 compared to clocks in the observer’s frame. The time dilation formula results in:

• \( \frac{1}{\sqrt{1 - \frac{3}{4}}} = 2 \)

The Lorentz factor, often denoted by \( \gamma \), is a key element in both length contraction and time dilation. It emerges as a result of transformations required to switch between different reference frames in special relativity.

• The Lorentz factor is expressed as: \( \gamma = \frac{1}{\sqrt{1 - \beta^2}} \)
• It tells us how much time, length, and relativistic mass change when moving at relativistic speeds.

For example, if you have a speed where \( \beta = \frac{\sqrt{3}}{2} \), the Lorentz factor would be:

• \( \gamma = \frac{1}{\sqrt{1 - \frac{3}{4}}} = 2 \)

This means that if you were on the spaceship, your time would "slow down" by a factor of 2 and the ship's length would appear contracted by that same factor, from the perspective of an outside observer.