The negative pion (π^-) is an unstable particle with an...

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?

Key Concepts

Special Relativity

Special relativity is a fundamental theory in physics that describes how measurements of time, length, and other physical properties differ for observers in different inertial frames of reference, especially at speeds close to the speed of light. This concept is crucial for understanding phenomena such as time dilation and length contraction when dealing with high-speed particles.

Time Dilation

Time dilation is a relativistic effect where the elapsed time between two events is longer for an observer in a different inertial frame compared to the proper time measured by a clock at rest relative to those events. This effect is quantified by the Lorentz factor and becomes significant at velocities approaching the speed of light.

Lorentz Factor

The Lorentz factor, often represented by the symbol gamma (?), is defined as 1/?(1 - v²/c²) and is a key component in the equations of special relativity. It relates the proper time and coordinate time, allowing calculations of time dilation and length contraction when transitioning between different inertial frames.

Proper Time

Proper time is the time interval measured by an observer at rest relative to the event being observed. It is considered the true time interval in the reference frame where the event occurs, and it is used as a baseline to compare with the dilated time measured in other frames.

Relativistic Kinematics

Relativistic kinematics involves the study of the motion of objects where velocities are close to the speed of light, and classical mechanics no longer applies. This field incorporates the effects of time dilation, length contraction, and the dependence of mass on velocity, which are all essential for correctly describing high-speed particles.

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