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Subatomic particles called pions are created when protons, acceler ated to speeds very near \(c\) in a particle accelerator, smash into th nucleus of a target atom. Charged pions are unstable particles the decay into muons with a half-life of \(1.8 \times 10^{-8}\) s. Pions have bee imvestigated for use in cancer treatment because they pass throug tissue doing minimal damage until they decay, releasing significar energy at that point. The speed of the pions can be adjusted so the the most likely place for the decay is in a tumor. Suppose pions are created in an accelerator, then directed int a medical bay \(30 \mathrm{m}\) away. The pions travel at the very high spee of \(0.99995 c .\) Without time dilation, half of the pions would hav decayed after traveling only \(5.4 \mathrm{m},\) not far enough to make it \(\mathrm{t}\) the medical bay. Time dilation allows them to survive long enoug to reach the medical bay, enter tissue, slow down, and then deca where they are needed, in a tumor. 81\. I What is the half-life of a pion in the reference frame of th patient undergoing pion therapy? A. \(1.8 \times 10^{-10} \mathrm{s}\) B. \(1.8 \times 10^{-8} \mathrm{s}\) C. \(1.8 \times 10^{-7} \mathrm{s}\) D. \(1.8 \times 10^{-6} \mathrm{s}\)

Step by step solution

01

Identify Given Values

From the problem, we know that the pion's half-life in its own frame (\(t\)) is \(1.8 \times 10^{-8} \) s, the speed of the pion (\(v\)) is \(0.99995 c\), and the speed of light (\(c\)) is the constant \(3.00 \times 10^8\) m/s.

02

Apply the Time Dilation Formula

We can now substitute these values into the time dilation formula to find \(t'\), the pion's half-life as observed by the patient: \(t' = t/\sqrt{1 - v^2/c^2}\).

03

Solve for \(t'\)

Substituting in the given values, you arrive at: \(t' = 1.8 \times 10^{-8} \mathrm{s} / \sqrt{1 - (0.99995)^2} = 1.8 \times 10^{-7} \mathrm{s}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Special Relativity

Special Relativity is a fundamental theory in physics that describes how objects behave at high speeds, especially those approaching the speed of light. Developed by Albert Einstein in 1905, this theory revolutionized how we understand time and space.
For objects moving very fast, like pions in a particle accelerator, the effects of special relativity become noticeable. One key concept is time dilation, where time appears to move slower for objects traveling at high speeds compared to a stationary observer. This means a fast-moving particle will experience less time passing than someone observing it from a standstill.
In our scenario, the pions' high velocity means they age slower from the patient's viewpoint. This allows them to travel 30 meters to the medical bay before decaying, rather than decaying too soon.

Particle Physics

Particle physics is the branch of physics that studies the smallest constituents of matter and energy interactions. These include subatomic particles like protons, neutrons, and pions. Pions, in this context, are important because they are produced in high-energy collisions, such as those created in particle accelerators.
Understanding these particles is crucial for applications ranging from particle therapy to fundamental research at facilities like CERN. Pions are particularly interesting in particle physics due to their properties and interactions, which include decaying into other particles like muons.
By studying pions' behaviors, scientists can harness their decay properly. This understanding also advances our knowledge of the universe, explaining fundamental forces and potential medical applications.

Pion Therapy

Pion therapy is an advanced form of cancer treatment that uses charged pions to target tumors. It leverages the physical properties of pions, which are unique because of their tendency to pass through healthy tissue with minimal damage. When they decay, they release energy, thereby delivering a precise dose of radiation to the tumor cells.
The advantage of this method is its precision. By adjusting the speed of pions, doctors can control where they decay, making it an effective therapy for deep-seated tumors. The surrounding healthy tissue receives low radiation, minimizing side effects.
In our exercise, the time dilation effect ensures the pions maintain their journey within the patient's body, increasing the chance of decaying within the tumor for optimal treatment results.

Half-life Concept

The half-life is a concept that describes the time required for half of a given quantity of a substance to decay. It's a critical measure in both physics and chemistry, especially for unstable particles like pions. They have a short half-life, meaning they decay quickly.
In particle physics, understanding the half-life of particles allows scientists to predict how they will behave over time. For pions, their intrinsic half-life is just 18 nanoseconds. However, under the influence of time dilation, this observed half-life extends as seen by a stationary observer.
This extended half-life becomes crucial in pion therapy, as it allows sufficient travel time for pions to reach tumors before decaying. Calculating these changes requires understanding the principles of special relativity to ensure that medical treatments become effective.