Some radioactive particles are traveling at \(0.999 c .\) If their lifetime is \(10^{-20}\) s when they are at rest, what is their lifetime at this speed? How far do they travel in that time (as viewed in the frame at which they are moving at \(0.999 c ?\)

Time dilation is a phenomenon predicted by Einstein's theory of Special Relativity. It describes how time can appear to slow down for an object moving at high speeds relative to an observer. The equation used to calculate this is:
\[ t = \frac{t_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \] where:

• \( t \) is the dilated time experienced by the moving object.
• \( t_0 \) is the proper time, the time experienced by the object at rest.
• \( v \) is the velocity of the moving object.
• \( c \) is the speed of light (approximately \(3\times10^8 \) meters per second).

When the velocity \( v \) of an object approaches the speed of light \( c \), the term \(\sqrt{1 - (v/c)^2}\) becomes very small. This causes the dilated time \( t \) to increase significantly. This effect is incredibly pronounced at speeds close to the speed of light, where even small increases in velocity lead to substantial time dilation.

Proper time, denoted by \( t_0 \), is the time interval measured by a clock that is at rest relative to the observer. In other words, it is the time experienced directly by an object or system. For example, if you are sitting at rest, the time you experience is your proper time.

In the context of the given exercise:

• The proper time \( t_0 \) for the radioactive particles at rest is given as \(10^{-20}\) seconds.

This forms the basis for calculating how their perceived lifetime changes when they move at high speeds. Proper time is always the shortest time interval between two events as measured in the frame of reference in which the events occur at the same location.

The speed of light, denoted by \( c \), is a fundamental constant in physics. It is approximately \(3 \times 10^8\) meters per second (\(m/s\)). It is essential in the study of relativity as it represents the maximum speed at which information or matter can travel.

In the formula for time dilation, \( c \) acts as a benchmark to compare the object's speed \( v \) against. When the object’s speed \( v \) approaches \( c \), time dilation effects become very significant.

• For particles moving at \(0.999c\), the dilated time is computed using this constant.

A notable property of the speed of light is its invariance; it remains constant in all frames of reference, regardless of the observer's relative motion. This unique characteristic is a cornerstone of Einstein’s Special Relativity.

In a relativistic frame, the distance traveled by an object moving at high speeds can be calculated by considering both its velocity and the dilated time experienced by that object.

Given in the exercise:

• Velocity \( v = 0.999c \)
• Dilated time \( t \approx 2.24 \times 10^{-19} \) seconds

The distance \( d \) traveled in the observer's frame can be calculated using the formula:
\[ d = v \times t \]

So, plugging in the known values:
\[ d = 0.999 c \times 2.24 \times 10^{-19} \approx 6.72 \times 10^{-11} \text{ meters} \]

This distance is the actual path length observed in the frame where the particles are moving at high speed. It vividly illustrates how even tiny time dilations can result in significant distances traveled for objects moving near the speed of light.