Problem 48 - Fundamentals of Physics

Question

The mass of a muon is 207 times the clectron mass; the average lifetime of muons at rest is \(2.20 \mu \mathrm{s}\). In a certain experiment, muons moving through a laboratory are measured to have an average lifetime of \(6.90 \mu \mathrm{s}\). For the moving muons, what are (a) \(\beta, (\) b) \(K,\) and \((c) p(\) in \(\mathrm{MeV} / c) ?\)

Step-by-Step Solution

Answer

(a) \(\beta \approx 0.948\), (b) \(K \approx 228.16\, \mathrm{MeV}\), (c) \(p \approx 317.33\, \mathrm{MeV}/c\)."

Step 1: Understanding the Problem

We are given the rest lifetime of muons, the moving lifetime, and need to find the velocity (\(\beta\)), kinetic energy (\(K\)), and momentum (\(p)\) of the moving muons. We use the time dilation formula and relativistic equations for energy and momentum.

Step 1: Calculate Velocity (Beta)

Using time dilation: \[ t' = \frac{t}{\sqrt{1-\beta^2}} \] where \(t = 2.20 \mu s\) (rest lifetime) and \(t' = 6.90 \mu s\) (moving lifetime). Solving for \(\beta\): \[ 6.90 = \frac{2.20}{\sqrt{1-\beta^2}} \] \[ \sqrt{1-\beta^2} = \frac{2.20}{6.90} \] \[ 1-\beta^2 = \left(\frac{2.20}{6.90}\right)^2 \] \[ \beta^2 = 1 - \left(\frac{2.20}{6.90}\right)^2 \] \[ \beta \approx \sqrt{0.8989} \approx 0.948 \]

Step 2: Calculate Kinetic Energy (K)

The relativistic kinetic energy is \[ K = (\gamma - 1)mc^2 \]. To find \(\gamma\): \[ \gamma = \frac{1}{\sqrt{1 - \beta^2}} \approx \frac{1}{0.3162} \approx 3.16 \]. The muon mass \(m = 207m_e\). \(mc^2 = 207 \times 0.511 \mathrm{MeV} = 105.777 \mathrm{MeV}\). Thus, \[ K = (3.16 - 1) \times 105.777 = 228.16 \mathrm{MeV} \].

Step 3: Calculate Momentum (p)

The relativistic momentum is \[ p = \gamma mv \] and since \( v = \beta c \), \[ p = \gamma m \beta c \]. With \(\beta = 0.948\) and \(\gamma = 3.16\), \[ p = 3.16 \times 207m_e \times 0.948c = (3.16 \times 0.948) \times 105.777 \mathrm{MeV}/c \]. Simplifying, \[ p \approx 3.00 \times 105.777 \approx 317.33 \mathrm{MeV}/c \].

Key Concepts

Time Dilation

In the realm of relativity in physics, time dilation is a fascinating phenomenon, emphasizing how time can differ for observers in various frames of motion. Essentially, it suggests that time ticks differently depending on how fast you are moving relative to another observer. In the context of muons in our original exercise, these particles exist at such high speeds that relativistic effects like time dilation become significant.

When muons are at rest, they have a specific lifetime, noted here as 2.20 microseconds. However, when racing through a laboratory with high speeds, observers measure a considerably longer lifetime of 6.90 microseconds. This phenomenon is explained by the time dilation formula:

• \[ t' = \frac{t}{\sqrt{1-\beta^2}} \]

where:

• \( t' \) is the observed time duration (6.90 microseconds)
• \( t \) is the proper time or rest time (2.20 microseconds)
• \( \beta \) is the velocity factor as a fraction of the speed of light \(c\).

The implication is that time, which appears invariant in our day-to-day life, stretches and contracts based on relative speeds, exemplified in high-speed muon experiments.

Relativistic Kinetic Energy

When objects move at speeds approaching that of light, traditional notions of kinetic energy change, giving rise to what we call relativistic kinetic energy. This form of kinetic energy accounts for the effects of relativity, being crucial for particles like muons traveling near the speed of light. Using the formula:

• \[ K = (\gamma - 1)mc^2 \]

relativistic effects are considered. Here:

• \( K \) is the relativistic kinetic energy.
• \( \gamma \) (gamma) represents the Lorentz factor, given by \( \frac{1}{\sqrt{1 - \beta^2}} \).
• \( m \) is the rest mass of the particle.
• \( c \) is the speed of light.

For our case with muons showcasing significant speed, the more substantial kinetic energy highlights how they store more energy than classical physics would predict. The mass energy equivalence, \( mc^2 \), often known as Einstein's relation, is the baseline from which we adjust our kinetic energy computation. Because of relativistic effects, as \( \beta \) approaches 1, \( \gamma \) increases dramatically, thus increasing the kinetic energy more than linearly, showcasing the stark departure from Newtonian mechanics.

Relativistic Momentum

Beyond kinetic energy, particles like muons also exhibit relativistic momentum when approaching light speed, differing from classical momentum understanding. Relativistic momentum is expressed as:

• \[ p = \gamma m v \]

where:

• \( p \) is the relativistic momentum.
• \( \gamma \) is the Lorentz factor, previously determined.
• \( m \) is the rest mass of the particle (in our case, for muons).
• \( v \) is its velocity, represented as \( \beta c \), where \( \beta \) is the velocity as a fraction of the speed of light.

Relativistic momentum captures how momentum increases with speed, emphasizing that momentum grows not only with mass and velocity but also with the factor \( \gamma \). Thus, when dealing with high-speed entities, their momentum becomes considerably larger than traditional linear equations would suggest.
In the researched scenario, the outcome indicates a significantly shifted momentum value of the muon, affirming how its velocity contributes dramatically to the overall momentum.

Muon Lifetime

Muons, subatomic particles with a charge similar to electrons, are known for their incredibly short lifetimes. At rest, they last approximately 2.20 microseconds, but when moving at high velocities, such as near the speed of light, they exhibit extended lifetimes due to relativistic effects, notable in scientific experiments. The extended lifetime of moving muons is directly linked to time dilation. As they fly through experimental setups at relativistic speeds, their lifespan stretches from what is measured in the stationary frame. Consequently, observers measure their lifetime as 6.90 microseconds, three times more than their rest lifetime.
Muon lifetime is critical in experimental physics, helping researchers confirm predictions of relativity and observe how time dilation works in real-world scenarios. These experiments provide insights, not just into how fundamental particles behave but also into verifying Einstein's theory, offering tangible evidence of time distortion under high velocity conditions.