Problem 1 - Fundamentals of Physics

Question

The mean lifetime of stationary muons is measured to be \(2.2000 \mu \mathrm{s}\). The mean lifetime of high-speed muons in a burst of cosmic rays observed from Earth is measured to be \(16.000 \mu \mathrm{s}\). To five significant figures, what is the speed parameter \(\beta\) of these cosmic-ray muons relative to Earth?

Step-by-Step Solution

Answer

The speed parameter \( \beta \) of cosmic-ray muons is approximately 0.99049.

Step 1: Identify Known Values

We know the mean lifetime of stationary muons is \( \tau_0 = 2.2000 \mu \mathrm{s} \) and the mean lifetime of high-speed muons is \( \tau = 16.000 \mu \mathrm{s} \).

Step 2: Use Time Dilation Formula

We use the time dilation formula from special relativity, \( \tau = \frac{\tau_0}{\sqrt{1-\beta^2}} \), where \( \beta \) is the speed parameter \( \frac{v}{c} \), \( v \) being the speed of the muons and \( c \) the speed of light.

Step 3: Rearrange the Formula for Beta

Rearrange the formula to solve for \( \beta \): \( \sqrt{1-\beta^2} = \frac{\tau_0}{\tau} \). This is further rearranged to \( 1 - \beta^2 = \left( \frac{\tau_0}{\tau} \right)^2 \).

Step 4: Calculate Beta

Calculate \( \beta^2 = 1 - \left( \frac{\tau_0}{\tau} \right)^2 \). Substituting the given values: \( \beta^2 = 1 - \left( \frac{2.2000}{16.000} \right)^2 = 1 - \left( 0.1375 \right)^2 = 1 - 0.01890625 = 0.98109375 \). Thus, \( \beta = \sqrt{0.98109375} = 0.9904918651 \).

Key Concepts

special relativity

Special relativity is a theory proposed by Albert Einstein that revolutionized our understanding of space, time, and energy. One of the central ideas of this theory is that the laws of physics are the same for all non-accelerating observers, no matter their velocity relative to one another. This was a groundbreaking change from Newtonian physics, which did not account for the effects of high speeds. In special relativity, time can slow down or "dilate," and lengths can contract depending on an object's speed relative to an observer. This is known as time dilation or length contraction. These effects only become noticeable at speeds approaching the speed of light. Special relativity also introduced the concept of mass-energy equivalence, summarized by the famous equation \(E = mc^2\), showing that mass can be converted into energy and vice versa.

speed of light

In the vacuum of space, the speed of light \(c\) is approximately \(299,792,458\) meters per second. It is the ultimate speed limit in the universe, meaning nothing with mass can travel at or above this speed. The speed of light plays a crucial role in special relativity, acting as a constant that forms the framework for measuring time and space. Einstein's theory suggests that as an object moves closer to the speed of light, time for that object will appear to slow down when observed from a stationary point. This is why high-speed particles, like the muons from cosmic rays, experience much longer lifetimes from our earthly perspective. It's crucial to note that the speed of light remains constant regardless of the observer's relative motion, making it fundamental to the laws of physics as we understand them.

muon decay

Muons are subatomic particles similar to electrons, but with a much greater mass. They are unstable and eventually decay into electrons and neutrinos. Stationary muons, those not in motion, have a mean lifetime of approximately \(2.2 \mu\text{s}\). However, when these particles are moving at speeds near that of light, their observed lifetime increases dramatically due to the effects of time dilation. In this context, high-speed muons generated by cosmic rays can have lifetimes measured in microseconds from our point of observation on Earth. This extended lifetime is a direct consequence of time dilation—a particle moving at high velocity relative to an observer experiences time more slowly. Hence, the high-speed muons created in the upper atmosphere due to cosmic ray interactions reach the Earth's surface without decaying as rapidly as they otherwise would.

cosmic rays

Cosmic rays are energetic particles originating from space that bombard the Earth. They are primarily composed of high-energy protons and atomic nuclei. When these cosmic rays collide with molecules in the Earth’s atmosphere, they produce a shower of secondary particles, including muons. Cosmic rays are a fascinating area of astrophysical study because they provide evidence of phenomena occurring in the distant universe, such as supernovae and other high-energy events. The study of cosmic rays and their interactions with Earth's atmosphere helps scientists understand more about fundamental particles and forces. In the atmosphere, the production of muons by cosmic rays presents a unique opportunity to see special relativity in action, specifically through the phenomenon of time dilation, as the muons created travel near the speed of light and thus have significantly extended lifetimes from our perspective.