Question

A pion is created in the higher reaches of the 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, pions have a lifetime of \(26 \mathrm{~ns}\). As measured in a frame fixed with respect to the Earth, how far will such a typical pion move through the atmosphere before it decays?

Short answer

The distance covered by a pion before it decays, as observed from Earth, is given by the time dilation multiplied by the speed of the pion. This value can be calculated by substituting the given values into the formula and calculating.

Step-by-step solution

Understand Time Dilation

The phenomenon of time dilation in special relativity tells us that the passage of time can seem slower to an observer if the observed object is moving close to the speed of light. In the given exercise, it is stated that a pion moves at a speed of 0.99 c, where c is the speed of light.

Calculate Time Dilation

The time dilation effect can be calculated using the formula \[ \Delta t = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}} \] where \(\Delta t\) is the time elapsed according to an Earth observer, \(\Delta t_0\) is the proper time (in this case, the lifetime of the pion which is 26 ns), and \(v\) is the velocity of the pion relative to the speed of light c. Plugging in the given values, we calculate the dilated time \[ \Delta t = \frac{26 \times 10^{-9} \,s}{\sqrt{1 - (0.99)^2}} \]

Calculate Distance Travelled

The distance travelled by the pion before it decays can be estimated using the formula for distance which is the speed of the pion multiplied by the time taken to decay as observed from Earth in the Earth's frame of reference, i.e., the time dilated value that we have calculated in the previous step. Thus, distance \( d = v \cdot \Delta t \)

Key concepts

Special Relativity

Imagine sitting in a train moving very fast and tossing a ball straight up. To you, the ball goes up and down vertically, but for an observer outside, the ball follows a longer, diagonal path. Yet both observations are correct; this is the essence of special relativity. Special relativity, a theory proposed by Albert Einstein, fundamentally changes our understanding of space and time.

According to special relativity, the laws of physics are the same for all non-accelerating observers, and the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer. One of the most fascinating consequences of this theory is time dilation, which states that time can pass at different rates for two observers based on their relative speeds.

For high-speed objects traveling near the speed of light, time appears to slow down when observed from a stationary frame of reference. The exercise about pions decaying in Earth's atmosphere provides a concrete example of this phenomenon: the moving pions' decay lifespan seems longer as observed from Earth compared to their own rest frame.

Cosmic Rays

Every day, Earth is bombarded by cosmic rays—high-energy particles from outer space with immense speeds. These particles are mostly protons and nuclei of various elements, with energies that can surpass anything we could artificially produce on Earth.

Cosmic rays can originate from our own sun, distant stars, or even galaxies outside the Milky Way. When these energetic particles enter Earth's atmosphere, they collide with atomic nuclei and lead to a cascade of secondary particles, including pions, as described in the exercise.

Interaction with the Atmosphere

In these collisions, energy converts to mass, giving birth to various subatomic particles. Pions created from such interactions inherit high velocities and play a significant role in the natural operating processes of our universe, offering insights into fundamental physics through their behavior and interactions.

Pions

Pions, or pi mesons, are subatomic particles that serve as one of the mediators of the strong nuclear force, which is the force that holds the nucleus of an atom together. They come in three forms: positively charged, negatively charged, and neutral.

Pions are not stable and have very short lifespans; they decay into other particles such as muons and neutrinos. The study of pions provides valuable information on the strong nuclear force and is crucial in understanding elementary particle physics. In the context of our exercise, the pions created from cosmic rays are traveling at a significant fraction of the speed of light and experience time dilation, which allows them to travel further in Earth's atmosphere before they decay.

Speed of Light

The speed of light is not just a fascinating speed limit; it is a fundamental constant of the universe. In vacuum, light travels at approximately 299,792 kilometers per second (or about 186,282 miles per second). This speed is denoted by the letter 'c' in equations and is the same no matter the frame of reference.

It's important in many areas of physics, including special relativity, where it factors into the equations that describe how measurements of space and time differ for observers in different inertial frames of reference. For instance, in the given textbook exercise, the pion's speed is given as a proportion of the speed of light, and this is key to calculating time dilation and ultimately, how far the pion travels before decaying.