Question

The positive muon (\(\mu^+\)), an unstable particle, lives on average 2.20 \(\times\) 10\(^{-6}\) 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 0.900c, what average lifetime is measured in the laboratory? (b) What average distance, measured in the laboratory, does the particle move before decaying?

Short answer

(a) Laboratory lifetime is 5.05 \( \times \) 10\(^{-6}\) seconds. (b) Distance traveled is 1.36 \( \times \) 10\(^3\) meters.

Step-by-step solution

Understand Time Dilation

The lifetime of a moving particle as measured in the laboratory frame differs from the lifetime measured in its own frame due to time dilation, a consequence of Einstein’s theory of relativity. The time dilation formula is \( t = \frac{t_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \), where \( t_0 \) is the proper lifetime, \( v \) is the speed of the particle, and \( c \) is the speed of light.

Calculate the Dilated Lifetime

Using the time dilation formula, substitute \( t_0 = 2.20 \times 10^{-6} \) s, \( v = 0.900c \), and \( c \) as the speed of light. The laboratory lifetime \( t \) is given by \[ t = \frac{2.20 \times 10^{-6}}{\sqrt{1 - (0.900)^2}}. \]Calculate the square root term and complete the division to find \( t \).

Plug in Values and Simplify

Calculate \( 1 - (0.900)^2 = 1 - 0.81 = 0.19 \). Consequently, \( \sqrt{0.19} \approx 0.436 \). Substitute this result back into the equation: \[ t = \frac{2.20 \times 10^{-6}}{0.436}. \]Divide to find \( t \approx 5.05 \times 10^{-6} \) seconds.

Calculate the Distance Traveled

In the laboratory frame, the distance \( d \) the particle travels is given by \( d = v \times t \). Substitute \( v = 0.900c \) and \( t = 5.05 \times 10^{-6} \) s. The distance \( d \) is \[ d = 0.900 \times 3 \times 10^8 \times 5.05 \times 10^{-6}. \]Calculate this multiplication to find \( d \).

Final Calculation and Solution

Calculate \( d = 0.900 \times 3 \times 10^8 \times 5.05 \times 10^{-6} = 1.36 \times 10^3 \) meters. This is the average distance the muon travels before decaying as measured in the laboratory.

Key concepts

Relativistic Effects

Relativistic effects are fascinating phenomena that arise due to observing events at speeds approaching the speed of light. One such effect is time dilation, which plays a pivotal role in modern physics. When particles like muons travel at velocities close to the speed of light, their behaviors contrast significantly with classical expectations. This is because relativistic effects cause time to stretch from the perspective of an external observer. A moving clock ticks slower compared to a stationary clock due to this phenomenon.

• Objects moving near light speed experience changes in mass, time, and length.
• Time dilation is a key factor in understanding these changes.
• The formula for time dilation involves the speed of the particle relative to the speed of light.

Time dilation allows us to calculate how long a particle's lifetime appears from the lab compared to being at rest, which is foundational in understanding high-speed particle behaviors.

Particle Lifetime

The lifespan of particles, such as muons, greatly impacts their study in physics. Measured in their rest frame, the lifetime of a positive muon is an average of 2.20 microseconds. However, this lifetime increases when observed from a stationary frame because of time dilation.

• At rest, the muon's average lifetime is exactly 2.20 microseconds.
• When moving, the lifetime is perceived as longer by observers.
• This dilation is a result of the particle's speed approaching light speed.

Understanding particle lifetime requires recognizing that these measurements differ based on the observer’s frame of reference. This change in perception helps scientists predict how far particles can travel before decaying, aiding in experimental physics.

Muon Decay

Muon decay is a process in which a muon, an elementary particle similar to the electron, transforms into other particles. This typically involves the conversion into an electron, a neutrino, and an antineutrino. The decay process helps physicists understand particle interactions and the conservation laws in particle physics.

• Muons are heavier cousins of electrons and unstable.
• Understanding their decay properties lets physicists explore fundamental forces.
• Measuring travel distance and decay helps in exploring relativistic effects.

The study of muon decay processes is critical in experimental physics, particularly in understanding how conditions like near-light speeds influence particle behavior and decay pathways.

Einstein's Theory of Relativity

Einstein's theory of relativity is a groundbreaking concept that revolutionized our understanding of space, time, and energy. It comprises the special and general theories of relativity, with the former particularly pertinent to inertial frames of reference and the latter to gravitational fields.

• Special relativity describes how time and space are interconnected.
• General relativity extends these concepts to include gravity's influence.
• Special relativity predicts time dilation and mass increase at high speeds.

The theory provides the framework for analyzing particle dynamics at high speeds. For example, Einstein’s equations help us understand how particles like muons behave when moving close to the speed of light, explaining why they appear to live longer than they do at rest. This theory underscores nearly all modern physics pertaining to fast-moving objects and their observed behaviors.