After being produced in a collision between elementary...

After being produced in a collision between elementary particles, a positive pion ($\pi^+$) must travel down a 1.90-km-long tube to reach an experimental area. A $\pi^+$ particle has an average lifetime (measured in its rest frame) of 2.60 $\times$ 10$^{-8}$ s; the $\pi^+$ we are considering has this lifetime. (a) How fast must the $\pi^+$ travel if it is not to decay before it reaches the end of the tube? (Since $u$ will be very close to $c$, write $u$ = (1 - $\Delta$)c and give your answer in terms of $\Delta$ rather than $u$.) (b) The $\pi^+$ has a rest energy of 139.6 MeV. What is the total energy of the $\pi^+$ at the speed calculated in part (a)?

Key Concepts

Rest Energy

Rest energy is the energy that a particle possesses due to its mass when it is not moving, encapsulated by the famous equation E = mc². This intrinsic energy is a baseline over which additional energy due to motion (kinetic energy) is added in relativistic contexts.

Relativistic Energy

Relativistic energy refers to the total energy of an object moving at speeds comparable to the speed of light. It is given by the formula E = ?E?, where E? is the particle's rest energy. This concept is crucial because, as a particle's speed approaches c, its kinetic energy increases dramatically due to the increase in the Lorentz factor.

Speed Representation Near c

In high-energy physics problems where particles travel extremely close to the speed of light, it is common to express the velocity in the form u = (1 - ?)c, where ? is a small number representing the deviation from the speed of light. This representation simplifies calculations by explicitly acknowledging that the speed is nearly c, while still accounting for minute differences essential to precision in relativistic computations.

Time Dilation

Time dilation is a fundamental concept in special relativity that explains how the passage of time is relative depending on the observer's frame of reference. For a moving particle, its decay lifetime measured in the laboratory frame is longer than its proper lifetime (i.e., the lifetime measured in its rest frame) by a factor of the Lorentz factor. This effect is critical in analyzing how fast the particle must travel to cover a specific distance before decaying.

Lorentz Factor

The Lorentz factor, typically denoted by gamma (?), is defined by the expression ? = 1/sqrt(1 - (u/c)^2), where u is the velocity of the particle and c is the speed of light. This factor quantifies the effects of time dilation and length contraction. It is essential in calculating how moving particles experience time and energy relative to stationary observers.

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