Problem 2 - University Physics with Modern Physics

Question

The positive muon \(\left(\mu^{+}\right),\) an unstable particle, lives on average \(2.20 \times 10^{-6} \mathrm{s}\) (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of \(0.900 c,\) what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

Step-by-Step Solution

Answer

(a) The average lifetime is approximately 5.05 µs. (b) The muon travels about 1364 meters.

Step 1: Understanding Time Dilation

In the lab frame, the moving muon experiences time dilation, so its lifetime is longer than in its rest frame. The formula for time dilation is given by: \[ t' = \frac{t_0}{\sqrt{1 - v^2/c^2}} \] where \( t_0 \) is the proper time, \( v = 0.900c \) is the speed, and \( c \) is the speed of light.

Step 2: Calculate Time Dilation

Given \( t_0 = 2.20 \times 10^{-6} \) s and \( v = 0.900c \), we can calculate the dilated lifetime in the lab frame:\[ t' = \frac{2.20 \times 10^{-6}}{\sqrt{1 - (0.900)^2}} \]Calculate the denominator: \[ \sqrt{1 - (0.900)^2} = \sqrt{1 - 0.81} = \sqrt{0.19} \approx 0.4359 \] Thus,\[ t' = \frac{2.20 \times 10^{-6}}{0.4359} \] \[ t' \approx 5.05 \times 10^{-6} \text{ s} \]

Step 3: Calculate Average Distance

The average distance the muon travels in the lab frame before decaying is given by: \[ d = v \cdot t' \] We already calculated \( t' \) as \( 5.05 \times 10^{-6} \text{ s} \). Now we find \( d \):\[ d = 0.900c \cdot 5.05 \times 10^{-6} \]\[ c = 3 \times 10^8 \text{ m/s} \] \[ d = 0.900 \times 3 \times 10^8 \times 5.05 \times 10^{-6} \]\[ d \approx 1.364 \times 10^3 \text{ m} \]

Key Concepts

Muon Decay

Muons are fascinating particles that are similar to electrons but much heavier. They belong to a family of particles called lepton. One big difference between a muon and an electron is that muons are unstable. This means they don't last very long before they transform or "decay" into other particles. Their average lifespan is measured as the time it takes for half of a large group of identical muons to decay.
In particle physics, understanding how long a particle lives is crucial. For muons, this lifetime is about 2.20 microseconds. This lifetime is the "proper time," measured in the particle's own frame of reference.
Using this decay property, scientists can learn more about the fundamental forces in nature and the properties of the particles involved. Studying muon decay also helps to verify various principles of physics, such as time dilation from special relativity.

Relativistic Effects

When particles like muons travel at high speeds, close to the speed of light, they exhibit relativistic effects. One such effect is known as time dilation. According to Einstein's theory of relativity, time does not behave the same way for objects moving at different speeds. Time appears to 'slow down' for an object moving close to the speed of light compared to an observer who is at rest.
This means that if a muon is moving at 90% the speed of light, it experiences more 'time' before decaying than if it were at rest. For observers on Earth, they would measure the muon's lifetime as longer than its proper time. This effect is calculated using the following formula:
\[ t' = \frac{t_0}{\sqrt{1 - v^2/c^2}} \]
where \( t' \) is the dilated time, \( t_0 \) is the proper time, and \( v \) is the muon's speed. As you can see, the closer \( v \) gets to the speed of light \( c \), the greater the dilation effect.

Proper Time

Proper time is an essential concept in relativity, referring to the time interval measured by a clock that is at rest relative to whatever event is being timed. For muons, this is the time that passes according to a clock moving along with the muon.
In relativistic physics, proper time is different from coordinate time, which is the time interval measured by an observer who is not at rest with respect to the event. The proper time for a muon is 2.20 microseconds, its lifespan when it is not moving relative to the clock measuring it.
Understanding proper time is crucial for calculating how time appears to pass differently from various observers' perspectives. It ensures we correctly apply time dilation when dealing with objects moving at significant fractions of the speed of light.

Particle Physics

Particle physics is a branch of physics that studies the fundamental constituents of matter and their interactions. It explores the tiniest bits of our universe, such as quarks and leptons, building blocks that make up protons, electrons, muons, and other particles.
In the realm of particle physics, muons play an important role. Despite their brief existence, they are key in experiments and research. Their interactions help physicists probe the mysteries of the universe. By studying particles like muons and their decay patterns, scientists can test and validate theoretical predictions in quantum mechanics and the standard model of particle physics.
This field involves complex theories, but its purpose is quite a simple one: to understand the most basic elements of nature and the forces acting upon them. By doing so, it lays the groundwork for further advancements in technology and fundamental science.