2. if the pions are moving at that speed, what time passes for the pions instead of (2.6 \times 10^{-8}\text{ s})?

$\frac{v}{c} = 0.923c$

Question

2. if the pions are moving at that speed, what time passes for the pions instead of (2.6 \times 10^{-8}\text{ s})?

$\frac{v}{c} = 0.923c$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Time Dilation", "Special Relativity" ], "new_concepts": [], "current_concepts": [ "Time Dilation", "Special Relativity" ] } ## Identify given values and target unknown Using the *Special Relativity* knowledge point $$ \begin{aligned} &v = 0.923c \ &\Delta t = 2.6 \times 10^{-8}\text{ s} \ &\text{Target: Proper time } \Delta t_0 \end{aligned} $$ ## Calculate the Lorentz factor Using the *Special Relativity* knowledge point $$ \begin{aligned} &\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \ &\gamma = \frac{1}{\sqrt{1 - (0.923)^2}} \ &\gamma = \frac{1}{\sqrt{1 - 0.851929}} \approx 2.5987 \end{aligned} $$ ## Calculate the proper time for the pions Using the *Time Dilation* knowledge point $$ \begin{aligned} &\Delta t = \gamma \Delta t_0 \ &\Delta t_0 = \frac{\Delta t}{\gamma} \ &\Delta t_0 = \frac{2.6 \times 10^{-8}\text{ s}}{2.5987} \approx 1.0 \times 10^{-8}\text{ s} \end{aligned} $$ $1.0 \times 10^{-8}\text{ s}$ { "subject": "Natural Science", "question_type": "Free Response", "knowledge_point": [ "Natural Science", "Physics", "Time Dilation" ] }

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Question

Explanation:

Identify given values and target unknown

Calculate the Lorentz factor

Calculate the proper time for the pions

Answer: