Question

the mean lifetime of certain subatomic particles is measured to be 2.3405 microseconds when they are stationary. however, the mena lifetime of fast-moving particles of the same kind observed in a burst of cosmic rays is measured to be 16.063 microsecond. what is the speed parameter of these cosmic-ray particles relative to earth?

Answer 1

To find the speed parameter of cosmic-ray particles relative to Earth, one can use the time dilation equation from special relativity. By substituting the observed and rest lifetimes into the formula and solving for the velocity v, one obtains the speed expressed as a fraction or percentage of the speed of light (c).

Step-by-step explanation:

The question posed involves calculating the speed of cosmic-ray particles relative to Earth based on the observed change in their mean lifetime, a concept rooted in special relativity. To solve for the speed parameter, also known as the velocity as a fraction of the speed of light (c), we will utilize the time dilation formula provided by Einstein's theory:

Lifetime observed = Lifetime at rest / sqrt(1 - (v^2/c^2)

Here, 16.063 microseconds is the lifetime of the particles as measured from Earth, and 2.3405 microseconds is the lifetime when the particles are at rest. Plugging the numbers in and solving for v will yield:

v = c * sqrt(1 - (2.3405 / 16.063)^2)

Given that c is the speed of light (approximately 3.00 x 10^8 m/s), we can find the value of v and express it as a fraction or a percentage of the speed of light. This will give us the speed parameter.