The positive muon (µ+), an unstable particle, lives on average 2.20 * 10^-6 s (measured in its own frame of reference) before decaying. (a) If such a particle is moving, with respect to the laboratory, with a speed of 0.900c, what average lifetime is measured in the laboratory?

Verified step by step guidance

1. Identify the given values: the proper lifetime of the muon (\(\tau_0\)) is 2.20 \(\times\) 10^-6 s, and the speed of the muon (\(v\)) is 0.900c, where c is the speed of light.
2. Recognize that the problem involves time dilation, a concept from the theory of special relativity, which describes how the time measured in different frames of reference can differ when objects are moving at speeds close to the speed of light.
3. Use the time dilation formula: \(\tau = \frac{\tau_0}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(\tau\) is the dilated lifetime (time measured in the laboratory frame) and \(\tau_0\) is the proper lifetime (time measured in the muon's rest frame).
4. Substitute the values into the time dilation formula: \(\tau = \frac{2.20 \times 10^{-6}}{\sqrt{1 - (0.900)^2}}\).
5. Calculate the value inside the square root, then compute the square root, and finally divide \(\tau_0\) by this value to find the dilated lifetime \(\tau\), which is the average lifetime of the muon as measured in the laboratory.

Key Concepts

Time Dilation

Time dilation is a phenomenon predicted by Einstein's theory of relativity, where time measured in a moving frame is longer than time measured in a stationary frame. For an observer in the laboratory, the lifetime of a moving particle appears extended due to its high velocity, which is a consequence of the Lorentz transformation. This effect becomes significant as the speed of the particle approaches the speed of light.

Lorentz Factor

The Lorentz factor, denoted as γ (gamma), quantifies the amount of time dilation experienced by an object moving at a significant fraction of the speed of light. It is calculated using the formula γ = 1 / √(1 - v²/c²), where v is the object's velocity and c is the speed of light. This factor is crucial for determining how much longer the lifetime of the muon will appear to an observer in the laboratory frame.

Rest Frame vs. Laboratory Frame

The rest frame of a particle is the frame of reference in which the particle is at rest, while the laboratory frame is the frame in which the particle is moving. Measurements of time, length, and other physical quantities can differ between these frames due to relativistic effects. Understanding the distinction between these frames is essential for correctly applying the principles of relativity to calculate the observed lifetime of the muon in the laboratory.