Question

The negative pion (${\pi }^{-}$) is an unstable particle with an average lifetime of 2.60 $\times$ 10${}^{-8}$s (measured in the rest frame of the pion). (a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be 4.20 $\times$ 10${}^{-7}$ s. Calculate the speed of the pion expressed as a fraction of c. (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

Short answer

The speed of the pion is approximately 0.9981c, and it travels about 125.77 meters in the laboratory.

Step-by-step solution

Understanding the Problem

We are given the lifetime of a pion in its rest frame as 2.60 $\times$ 10${}^{-8}$ s and its lifetime in the laboratory frame as 4.20 $\times$ 10${}^{-7}$ s. We need to find the speed of the pion as a fraction of the speed of light, $c$, using the time dilation effect, and calculate the distance it travels in the laboratory frame.

Applying Time Dilation Formula

To find the speed of the pion, we use the time dilation formula from special relativity: $\mathrm{\Delta }t=\mathrm{\Delta }{t}_0/\sqrt{1-v^2/c^2}$, where $\mathrm{\Delta }{t}_0=2.60\times {10}^{-8}$ s is the proper lifetime, $\mathrm{\Delta }t=4.20\times {10}^{-7}$ s is the dilated lifetime, and $v$ is the speed of the pion. Rearranging the formula to solve for $v$, we find that $v=c\sqrt{1-(\mathrm{\Delta }{t}_0/\mathrm{\Delta }t{)}^2}$.

Calculating the Speed

Substitute the given values into the formula: $v=c\sqrt{1-(2.60\times {10}^{-8}/4.20\times {10}^{-7}{)}^2}$ to calculate $v/c$. Simplify inside the square root: $v=c\sqrt{1-(0.0619{)}^2}$. So, $v=c\sqrt{1-0.0038}$ which approximately equals $v=c\sqrt{0.9962}$. Calculating, we get $v\approx 0.9981c$.

Calculating Distance Traveled

The distance traveled in the laboratory frame is $d=v\mathrm{\Delta }t$. Substituting the known values, $d=0.9981c\times 4.20\times {10}^{-7}$ s. Using $c=3.00\times {10}^8$ m/s, $d=0.9981\times 3.00\times {10}^8\times 4.20\times {10}^{-7}$. Simplifying gives $d\approx 125.77$ m.

Key concepts

special relativity

Special relativity, formulated by Albert Einstein, is a fundamental theory in physics that describes the physics of moving objects at high speeds, particularly those approaching the speed of light. It's grounded in two key postulates:

• The laws of physics are identical for all observers in uniform motion relative to one another (principle of relativity).
• The speed of light is the same for all observers, regardless of their motion relative to the light source.

This theory leads to some counterintuitive phenomena like time dilation and length contraction.
Time dilation, a primary concept within this framework, reveals that a moving clock ticks slower compared to a stationary clock.
For instance, a fast-moving pion lives longer when observed from a stationary laboratory frame than it does in its own rest frame.
As seen in the problem, the pion's lifetime extends when moving at high velocity, a direct demonstration of time dilation effects.

pion lifetime

The pion, specifically the negative pion (${\pi }^{-}$), is an unstable subatomic particle, often studied in particle physics. Pions are responsible for mediating the strong nuclear force between nucleons and are crucial in understanding nuclear interactions.
The lifetime of such particles, often very brief, depends heavily on their speed and the reference frame from which observations are made.
In its rest frame, a pion's average lifetime is measured as a fixed constant, in this case, 2.60 $\times$ 10${}^{-8}$ seconds.

• This value is based on decay processes intrinsic to the pion, unaffected by external factors like velocity.
• Understanding the lifetime of pions helps physicists probe deeper into the structure of matter at its most fundamental level.

The problem highlights how the pion's lifetime varies with its speed when viewed from a different frame, exemplifying time dilation in special relativity.

speed of light

The speed of light, denoted as $c$, is a fundamental constant essential in the study of physics. It is approximately equal to 3.00 $\times$ 10${}^8$ meters per second.

• This constant serves as the ultimate speed limit of the universe, meaning nothing with mass can travel faster than this speed.
• Light's consistent speed is central to many foundational principles, including special relativity.

In the exercise, calculating the pion's speed as a fraction of $c$ demonstrates how this constant is used to understand high-speed phenomena.
By applying the time dilation formula, the exercise determines the pion travels at about 0.9981 of the speed of light. This emphasizes how even small deviations in speed near $c$ have significant impacts on physics quantities like time and distance.

proper lifetime

In special relativity, the concept of proper lifetime is key to understanding how time is measured differently in various frames of reference.
The proper lifetime of a particle, like the pion in the given problem, is the time interval measured in the rest frame of the particle where it is not moving.

• In this frame, the pion's natural decay process proceeds at its normal rate.
• The proper lifetime is not affected by the particle's velocity or the observer's frame of reference.

In the exercise, the given proper lifetime of the pion is 2.60 $\times$ 10${}^{-8}$ seconds.
When the pion is observed from a laboratory frame, moving at a high speed, its lifetime appears to increase, illustrating time dilation effects.