Problem 45 - Fundamentals of Physics

Question

In a high-energy collision between a cosmic-ray particle and a particle ncar the top of Earth's atmosphere, \(120 \mathrm {~km}\) above sca level, a pion is created. The pion has a total energy \(E\) of \(1.35 \times 10^{5} \mathrm{MeV}\) and is traveling vertically downward. In the pion's rest frame, the pion decays \(35.0 \mathrm{~ns}\) after its creation. At what altitude above sea level, as measured from Earth's reference frame. does the decay occur? The rest cnergy of a pion is \(139.6 \mathrm{MeV}\).

Step-by-Step Solution

Answer

The decay occurs at approximately 109.865 km above sea level.

Step 1: Understand the Problem

We are given a pion created 120 km above sea level with a total energy of \(1.35 \times 10^5\) MeV and a pion rest frame decay time of 35.0 ns. We need to find out at what altitude the pion decays as measured from Earth's frame. This involves calculating the time dilation and using the speed of the pion.

Step 2: Calculate the Lorentz Factor (γ)

The Lorentz factor is defined as \( \gamma = \frac{E_{total}}{E_{rest}} \). \( E_{total} = 1.35 \times 10^5 \text{ MeV} \) and \( E_{rest} = 139.6 \text{ MeV} \). So, \( \gamma = \frac{1.35 \times 10^5}{139.6} \approx 967 \).

Step 3: Calculate Time Dilation in Earth's Frame

The time experienced in Earth's reference frame is \( t = \gamma \times t_{0} \) where \( t_{0} = 35.0 \text{ ns} \) is the proper time. So, \( t = 967 \times 35.0 \approx 33,845 \text{ ns} \).

Step 4: Calculate the Speed of the Pion

Since \( \beta = \frac{v}{c} \) and \( \gamma = \frac{1}{\sqrt{1 - \beta^2}} \), solve \( \beta \) using \( \gamma \).\ First, solve for \( \beta^2 \): \( 1 - \beta^2 = \frac{1}{\gamma^2} \ \beta = \sqrt{1 - \frac{1}{967^2}} \approx 0.9995 \). Thus, \( v \approx 0.9995c \).

Step 5: Calculate the Distance Traveled by the Pion

In Earth's frame, distance is calculated as \( d = v \times t \). Convert \( t = 33,845 \text{ ns} \) to seconds: \( t = 33,845 \times 10^{-9} \text{ s} \). So \( d = 0.9995 \times 3 \times 10^8 \times 33,845 \times 10^{-9} \approx 10,135 \text{ m} \approx 10.135 \text{ km} \).

Step 6: Calculate the Altitude at Which the Decay Occurs

The initial altitude is 120 km. Subtract the distance the pion travels: \ \( \text{Altitude of decay} = 120 - 10.135 = 109.865 \text{ km} \).

Key Concepts

Cosmic-ray collision

Cosmic-ray collisions occur when high-energy particles from space, such as cosmic rays, strike particles in the Earth's atmosphere. These collisions are crucial for understanding cosmic phenomena since they trigger further reactions.

• Cosmic rays mostly consist of high-energy protons or atomic nuclei that journey through space.
• When these rays collide with atmospheric particles, they generate secondary particles such as pions.
• Due to their high energy, cosmic-ray collisions play an important role in high-energy physics explorations.

Such a collision serves as the setting for our exercise, where a pion, a type of meson, was produced."},{