Problem 74 - Fundamentals of Physics

Question

A pion is created in the higher reaches of Earth's atmosphere when an incoming high-energy cosmic-ray particle collides with an atomic nucleus. A pion so formed descends toward Earth with a speed of \(0.99 c .\) In a reference frame in which they are at rest, pionsdecay with an average life of 26 ns. As measured in a frame fixed with respect to Earth, how far (on the average) will such a pion move through the atmosphere before it decays?

Step-by-Step Solution

Answer

The pion travels approximately 54.8 meters before decaying.

Step 1: Identify Relevant Concepts

In this problem, we need to apply concepts from special relativity, specifically time dilation, to determine the distance traveled by the pion. The average lifespan of the pion in its rest frame and its speed relative to Earth are given.

Step 2: Calculate Time Dilation

In the Earth frame, the life time of the pion is longer due to time dilation. The time dilated lifetime \( t' \) can be calculated using the formula: \( t' = \frac{t}{\sqrt{1-v^2/c^2}} \), where \( t = 26 \) ns is the rest lifetime, \( v = 0.99c \), and \( c \) is the speed of light.

Step 3: Solve for Dilated Time

Substitute the given values into the time dilation formula: \( t' = \frac{26}{\sqrt{1-(0.99)^2}} \). Calculating the square root and simplifying, we find \( \sqrt{1-(0.99)^2} = 0.141 \), so that \( t' = \frac{26}{0.141} \approx 184.4 \) ns.

Step 4: Calculate Distance Traveled

Determine the distance traveled by the pion using the formula \( d = v t' \). Here, \( v = 0.99c \) and \( t' \approx 184.4 \) ns. Use \( c = 3 \times 10^8 \) m/s for calculations: \( d = 0.99 \times 3 \times 10^8 \times 184.4 \times 10^{-9} \).

Step 5: Perform Distance Calculation

Calculate \( d = 0.99 \times 3 \times 10^8 \times 184.4 \times 10^{-9} \approx 54.8 \) meters. This is the average distance a pion travels before decaying in Earth’s atmosphere.

Key Concepts

Time Dilation

Time dilation is a key concept in Einstein's theory of special relativity. It describes how time passes at different rates in different reference frames. For example, when objects move at speeds close to the speed of light, their perception of time slows down compared to an observer at rest.

• According to special relativity, if two observers are moving relative to each other, each will see the other's clock as ticking more slowly.
• This effect becomes significant only at relativistic speeds, meaning speeds close to the speed of light.

In the context of the provided problem, a pion is moving at a speed of 0.99 times the speed of light. Its ability to travel further before decaying than its rest-life suggests, is due to time dilation.
For an observer on Earth, the pion's decay clock ticks slower, hence its lifespan is longer compared to its rest frame.

Cosmic Ray Particles

Cosmic ray particles are high-energy particles originating from outer space. These particles travel through the universe at nearly the speed of light.

• They come from various sources, such as the sun, distant stars, or even supernovae explosions.
• Upon entering Earth's atmosphere, cosmic rays collide with atmospheric particles, creating secondary particles such as pions.

Understanding the interaction of cosmic ray particles with Earth's atmosphere helps explain phenomena such as pion creation. This is a crucial element in studies related to particle physics and atmospheric science.
These interactions showcase the fascinating interplay between high-energy astrophysical processes and Earth's atmospheric layers.

Pion Decay

Pion decay is the process by which a pion transforms into other particles, releasing energy as it decays.

• Pions are unstable subatomic particles with a very short average lifespan, typically measured in nanoseconds.
• In its rest frame, a pion usually decays within 26 nanoseconds. However, this time extends when viewed from a frame where the pion is in motion, due to time dilation.
• The decay products of a pion can include neutrinos and, depending on the type of pion, other particles like muons.

This rapid decay is crucial in particle physics experiments, providing understanding of fundamental forces and symmetries in nature.
The analysis of pion decay helps physicists explore the weak nuclear force, one of the four fundamental forces of nature.

Reference Frames

Reference frames are perspectives from which observations are made in physics.

• A reference frame could be stationary like the Earth's surface, or in motion, such as a pion racing through the atmosphere.
• Special relativity dictates that physical laws, such as those governing decay or time measurement, are the same in all inertial frames.

Understanding reference frames is crucial for solving physics problems involving relative motion.
In the context of the problem given, two reference frames are involved: the rest frame of the pion and the Earth’s frame. The difference between measurements in these frames is resolved through the principles of special relativity.
By considering both frames, one can accurately assess physical phenomena like the pion's travel and its decay time.