Problem 6 - College Physics

Question

\(\bullet\) The negative pion \(\left(\pi^{-}\right)\) is an unstable particle with an average lifetime of \(2.60 \times 10^{-8} \mathrm{s}\) (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, as measured in the laboratory, does the pion travel during its average lifetime?

Step-by-Step Solution

Answer

(a) The speed is approximately 0.99808c. (b) The pion travels about 125.8 meters.

Step 1: Understand the problem

The problem involves the concept of time dilation in special relativity. We have a pion with a rest lifetime of \(2.60 \times 10^{-8} \mathrm{s}\) and a measured lifetime in the laboratory of \(4.20 \times 10^{-7} \mathrm{s}\). We need to calculate its speed as a fraction of the speed of light \(c\), and the distance it travels in the laboratory frame.

Step 2: Apply time dilation formula

The time dilation formula is \(\Delta t = \gamma \Delta t_0\), where \(\Delta t\) is the dilated time, \(\Delta t_0\) is the proper time, and \(\gamma\) is the Lorentz factor given by \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\). We set \(\Delta t = 4.20 \times 10^{-7}\, \mathrm{s}\) and \(\Delta t_0 = 2.60 \times 10^{-8}\, \mathrm{s}\).

Step 3: Solve for Lorentz factor and speed

From the equation \(4.20 \times 10^{-7} = \gamma \times 2.60 \times 10^{-8}\), solve for \(\gamma\): \(\gamma = \frac{4.20 \times 10^{-7}}{2.60 \times 10^{-8}} \approx 16.15\). Use \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) to solve for \(v\): \(\sqrt{1 - \frac{v^2}{c^2}} = \frac{1}{16.15}\) leading to \(1 - \frac{v^2}{c^2} = 0.00384\) which gives \(\frac{v^2}{c^2} = 0.99616\). Thus, \(v = c \times \sqrt{0.99616} \approx 0.99808c\).

Step 4: Calculate distance traveled

Using \(d = vt\), where \(t\) is the time in the laboratory frame and \(v = 0.99808c\), calculate the distance: \(d = 0.99808c \times 4.20 \times 10^{-7}\, \mathrm{s}\). Substituting \(c = 3 \times 10^8 \mathrm{m/s}\), the distance is \(d = 0.99808 \times 3 \times 10^8 \times 4.20 \times 10^{-7} \approx 125.8\, \mathrm{m}\).

Key Concepts

Time Dilation in Special Relativity

Time dilation is a fascinating concept arising from Einstein's theory of special relativity. It describes how time can pass at different rates for observers in different frames of reference. For instance, a clock moving at a high speed relative to an observer will appear to tick slower. In simpler terms, from our perspective on Earth, an astronaut traveling near the speed of light would age more slowly than someone on Earth.

This effect is quantified using the time dilation formula: \[\Delta t = \gamma \Delta t_0\] where

• \(\Delta t\) is the time interval measured by an observer (dilated time).
• \(\Delta t_0\) is the proper time interval (time for a stationary observer).
• \(\gamma\) is the Lorentz factor.

In the context of the exercise, the proper time \(2.60 \times 10^{-8} \mathrm{s}\) is the lifetime of the pion when at rest. The dilated time \(4.20 \times 10^{-7} \mathrm{s}\) is what someone in the lab measures. This demonstrates that moving particles experience time differently.

Understanding the Lorentz Factor

The Lorentz factor \(\gamma\) plays a crucial role in calculating time dilation and other relativistic effects. It's expressed as:\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\] This factor determines how much time, length, and relativistic mass will change for an object moving at velocity \(v\) compared to the speed of light \(c\).

• A \(\gamma\) value of 1 implies no change as seen in everyday low-speed scenarios.
• As speed approaches \(c\), \(\gamma\) increases significantly, leading to noticeable effects such as time dilation.

In the example, we computed \(\gamma \approx 16.15\) for a pion traveling nearly at the speed of light. This high value indicates that the pion's time runs much slower than that witnessed in the lab. Notably, \(\gamma\) becomes significantly larger as \(v\) nears \(c\), making relativistic effects more pronounced.

Speed of Light as a Cosmic Speed Limit

The speed of light, denoted by \(c\), is roughly \(3 \times 10^8\) meters per second. It is the ultimate speed limit of the universe — nothing with mass can travel as fast as light.

When considering particles like the pion, which travel at relativistic speeds, we express their velocity \(v\) as a fraction of the speed of light. In the exercise, we calculated the pion's speed to be about \(0.99808c\), illustrating how relativistic speeds approach but never reach \(c\).

This cosmic speed limit arises from the nature of spacetime itself, as described by Einstein's theory. Light travels at this constant speed in a vacuum, unaffected by the motion of its source or the observer, a fundamental principle of relativity. This unique characteristic ensures consistent physics throughout the universe, regardless of the relative motion of observers.

Unstable Particles in Physics

Unstable particles are elementary entities that do not last indefinitely; instead, they decay into other particles. These particles, like the negative pion \(\pi^{-}\), demonstrate fascinating behaviors under relativistic conditions.

Negative pions have a short lifetime at rest but appear to last longer when observed moving at high speeds, an effect explained by time dilation. For physicists, studying unstable particles helps reveal the inner workings of fundamental forces and particles.

• The rest lifetime of the negative pion is \(2.60 \times 10^{-8} \mathrm{s}\) -- a fleeting existence.
• When moving at speeds like \(0.99808c\), the lifetime extends to \(4.20 \times 10^{-7} \mathrm{s}\) as measured in the lab frame.

Exploring these particles within a relativistic framework provides deep insights into particle physics and the standard model, enhancing our understanding of the universe's fundamental aspects.