One of the fastest cosmic rays ever observed had a speed of \(\left(1.0 \mathrm{minus} 10^{-20}\right) \times c .\) Assume that the cosmic ray and a photon left a source at the same instant. To a stationary observer, how far behind the photon would the cosmic ray be after traveling for 100 million years?

Relativistic speeds refer to velocities that are a significant fraction of the speed of light. When objects move at these speeds, classical mechanics no longer accurately describe their behavior. Instead, we use Einstein's theory of relativity. The cosmic ray in this problem travels at nearly the speed of light, specifically \(1 - 10^{-20}\times c\), where \(c\) is the speed of light. At such high velocities, the effects of relativity, such as time dilation and length contraction, become significant. These speeds cause time to slow down and distance to contract from the perspective of a stationary observer.

Photons are particles of light and always travel at the speed of light, which is approximately \(3 \times 10^8 \text{ meters/second}\). To find out how far a photon travels in a given time, you can use the formula: \[ d_{\text{photon}} = c \times t \]. In this problem, the time given is 100 million years. Since \( d = c \times t \), we calculate the distance by first converting years to seconds. There are about \(3.154 \times 10^7 \) seconds in a year, so the total travel time is \[ t = 100 \times 10^6 \times 3.154 \times 10^7 \text{ seconds} \]. Multiplying this by the speed of light gives us the distance traveled by the photon.

Time dilation is a concept from Einstein's theory of relativity that states time passes slower for an object in motion compared to a stationary observer. This effect becomes more noticeable as the object's speed approaches the speed of light. For the cosmic ray traveling nearly at light speed, time dilation means that the time experienced by the cosmic ray would be different from that experienced by an observer at rest. While the cosmic ray and photon leave their source simultaneously, the cosmic ray’s internal clock runs slower due to its high velocity. This relativistic effect doesn't directly affect the distance calculation in this specific problem but is essential for understanding relativistic motion over extended periods.

The speed of light \((c)\) is a fundamental constant of nature, valued at approximately \(3 \times 10^8 \text{ meters/second}\). It is the ultimate speed limit of the universe, meaning nothing can travel faster than it. In this problem, we compare the distances traveled by a photon and a cosmic ray. While the photon travels at \c\, the cosmic ray travels slightly slower at \( (1 - 10^{-20}) \times c \). This tiny difference accumulates over time. Given 100 million years, we can form equations that derive the distances traveled by both the photon and the cosmic ray. The difference between these distances tells us how far behind the cosmic ray is compared to the photon after such a long time. By plugging the values into our formulas \[ d_{\text{photon}} = c \times t \] and \[ d_{\text{cosmic\text{ray}}} = (1 - 10^{-20}) \times c \times t \], we see a slight difference that can be calculated accurately using high precision.