Problem 48 - Fundamentals of Physics

Question

The mass of a muon is 207 times the electron 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 \(\mathrm{MeV} / \mathrm{c}\) )?

Step-by-Step Solution

Answer

(a) \(\beta \approx 0.951\), (b) \(K \approx 225.77\text{ MeV}\), (c) \(p \approx 315.48\text{ MeV/c}\)

Step 1: Determine the Time Dilation Factor

The time dilation factor can be identified using the relationship between the proper time (rest frame) and the dilated time (moving frame):\[ \gamma = \frac{t}{t_0} = \frac{6.90 \mu s}{2.20 \mu s}\]Solving this gives us:\[ \gamma = 3.136\]

Step 2: Calculate Velocity Parameter \( \beta\)

The velocity parameter \( \beta \) is calculated using the Lorentz factor \( \gamma \):\[ \gamma = \frac{1}{\sqrt{1-\beta^2}}\]Rearranging and solving for \( \beta \), we have:\[ \beta = \sqrt{1 - \frac{1}{\gamma^2}} = \sqrt{1 - \frac{1}{(3.136)^2}}\]\[ \beta \approx 0.951\]

Step 3: Calculate Kinetic Energy \( K \)

Kinetic energy can be determined using the relativistic kinetic energy formula:\[ K = (\gamma - 1) m c^2\]The rest mass energies of the muon are:\[ m_0 c^2 = 207 \times 0.511 \text{ MeV} = 105.777 \text{ MeV}\]Thus:\[ K = (3.136 - 1) \times 105.777 \text{ MeV} = 225.77 \text{ MeV}\]

Step 4: Calculate the Momentum \( p \)

The momentum \( p \) of a relativistic particle is calculated using the expression:\[ p = \gamma m \beta c\]Substitute the given values:\[ p = 3.136 \times 105.777 \text{ MeV/c} \times 0.951 = 315.48 \text{ MeV/c}\]

Key Concepts

Muon Decay

Muons are elementary particles similar to electrons but with greater mass—207 times that of an electron. They have an average lifetime measured in their rest frame, which is about 2.20 microseconds (μs). When a muon decays, it transforms into other particles, typically an electron along with neutrinos. This decay process allows physicists to study the particle's properties more closely. Observing muon decay in experiments can reveal important information about subatomic particles and fundamental forces in nature. It is especially important to understand how moving muons behave, as it ties directly into phenomena observed in relativistic physics. Their decay times appear different to an observer who sees the muons in motion. This is due to the effects of time dilation, a concept of Einstein's Theory of Special Relativity, highlighting the fascinating interplay between mass, time, and velocity at relativistic speeds.

Time Dilation

Time dilation is a phenomenon predicted by Einstein’s Theory of Special Relativity. It states that time measured in a moving system appears longer, or "dilated," compared to a system at rest. This can be expressed with the time dilation factor, \( \gamma \), which relates proper time\( t_0 \) to dilated time \( t \). In the case of muons, even if their rest lifetime is 2.20 μs, when they move at high velocities, we measure their lifetime differently—like in this example where it's 6.90 μs. This is because time shifts become significant at speeds nearing the speed of light. Understanding time dilation helps explain why we see longer lifetimes for fast-moving particles. It fundamentally illustrates how the perception of time can change based on the relative velocity of the observer and the moving object.

Lorentz Factor

The Lorentz factor, denoted \( \gamma \), is a key component in relativistic physics. It quantifies the effects of time dilation and length contraction experienced by objects moving at relativistic speeds. The Lorentz factor is calculated as: \[ \gamma = \frac{1}{\sqrt{1 - \beta^2}} \], where \( \beta = \frac{v}{c} \), and \( v \) is the object's velocity and \( c \) is the speed of light. For muons, as seen in our example, \( \gamma \) was about 3.136, indicating significant relativistic effects due to their high velocity. The Lorentz factor is pivotal in calculating other relativistic quantities, such as momentum and energy. Its value shows how much time and space are "stretched," reflecting the extreme conditions under which relativistic particles like muons travel.

Relativistic Momentum

Relativistic momentum differs from classical momentum. As velocities approach the speed of light, the classical formula \( p = mv \) no longer suffices, and relativistic momentum must be used: \[ p = \gamma m \beta c \]. This formula includes the Lorentz factor, \( \gamma \), accounting for the increased effective mass of the object as its velocity increases. In our muon example, with \( \gamma = 3.136 \), the relativistic effects are profound. The momentum calculated accounts for the near-light-speed muon velocity, which significantly inflates its value compared to a non-relativistic approach. Understanding relativistic momentum is essential for physics studies where speeds are a significant fraction of light speed, ensuring calculations remain accurate and applicable under these extreme conditions.

Kinetic Energy

In relativistic physics, kinetic energy is not simply \( \frac{1}{2} mv^2 \) as in classical mechanics. Instead, it is calculated using:\[ K = (\gamma - 1) m c^2 \]. This formula reflects how kinetic energy changes at velocities close to the speed of light. For our muon, with its mass at rest energy (207 times the electron mass), the increase in kinetic energy is significant when \( \gamma \) reaches 3.136. This means its kinetic energy substantially exceeds what classical calculations would predict. Relativistic kinetic energy is essential for understanding particle collisions and accelerations in high-energy physics, impacting how we view interactions and transformations at subatomic levels. Using the relativistic formula affirms Einstein's insight into energy-mass equivalence and the dramatic energy scales involved at relativistic speeds.