Problem 5 - Fundamentals of Physics

Question

An unstable high-energy particle enters a detector and leaves a track of length \(1.05 \mathrm{~mm}\) before it decays. Its speed relative to the detector was \(0.992 c .\) What is its proper lifetime? That is, how long would the particle have lasted before decay had it been at rest with respect to the detector?

Step-by-Step Solution

Answer

The proper lifetime is approximately \(5.77 \times 10^{-13} \text{ s}\).

Step 1: Understanding Time Dilation

When a particle moves at relativistic speeds, time dilation must be considered. Time dilation is explained by Einstein's theory of relativity, which states that time observed in the particle's (proper) frame and the detector's (observer) frame are different.

Step 2: Identify Given Data

We are given: the track length \(L = 1.05 \text{ mm}\), speed \(v = 0.992c\), where \(c\) is the speed of light, and we need to find the proper lifetime \(\Delta t_0\) of the particle.

Step 3: Relate Travel Distance to Proper Time

The time observed by the detector can be calculated using the classical relation: \(t = \frac{L}{v}\), where \(t\) is the time observed by the detector.

Step 4: Apply Time Dilation Formula

The time dilation formula is \(t = \gamma \Delta t_0\), where \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\) is the Lorentz factor. We use this to relate the proper time \(\Delta t_0\) and the observed time \(t\).

Step 5: Calculate Lorentz Factor

\(\gamma = \frac{1}{\sqrt{1 - (0.992)^2}}\). Calculation yields \(\gamma \approx 6.125\).

Step 6: Calculate Observed Time

\(t = \frac{L}{v} = \frac{1.05 \times 10^{-3} \, \text{m}}{0.992 \times 3 \times 10^8 \, \text{m/s}}\). This yields \(t \approx 3.533 \times 10^{-12} \text{ s}\).

Step 7: Determine Proper Lifetime

Using \(t = \gamma \Delta t_0\), find \(\Delta t_0 = \frac{t}{\gamma}\). Applying the values: \(\Delta t_0 \approx \frac{3.533 \times 10^{-12} \text{ s}}{6.125}\), the proper lifetime \(\Delta t_0 \approx 5.77 \times 10^{-13} \text{ s}\).

Key Concepts

Time Dilation

Time dilation is a fascinating concept from Einstein's theory of relativity. It describes how time can pass at different rates for observers who are in different states of motion.
For a particle traveling at speeds close to the speed of light (relativistic speeds), this difference becomes significant. When observing such a particle move, time appears to stretch or "dilate" for an observer compared to someone moving with the particle.
Here's how it works:

• The observer at rest (like a scientist with a detector) will see the clock of a fast-moving particle ticking slower than their own clock.
• This means when the particle decays in the observer's time frame, it seems to have taken longer than the time in the particle's own rest frame.

This peculiar effect allows particles to live longer from the perspective of the observer compared to what their lifetime would be if they were not moving at relativistic speeds.

Proper Lifetime

The proper lifetime is the time measured for a particle in its own rest frame.
It answers the question of how long the particle naturally takes to decay when not moving relative to the observer. This is crucial to understand how time dilation affects observations.In essence, the proper lifetime is the "true" or "invariant" lifetime of the particle, independent of its speed relative to other observers.
For the unstable high-energy particle mentioned in the exercise, its proper lifetime is the duration it would last if it were at rest with respect to the detector.

• To find the proper lifetime, we use the time dilation relationship: \[\Delta t_0 = \frac{t}{\gamma}\]where \(t\) is the time observed in the detector's frame, and \(\gamma\) is the Lorentz factor.
• This gives us a corrected, shorter duration compared to what the observer measures because of time moving slower in the particle's fast-moving frame.

Lorentz Factor

The Lorentz factor, denoted as \(\gamma\), is a key element in relativistic physics. It quantifies how much time, length, and relativistic mass change for an object while it moves at significant fractions of the speed of light.
This factor is calculated using the formula:\[\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \]where \(v\) is the speed of the object and \(c\) is the speed of light.

• When the speed \(v\) is much less than the speed of light, \(\gamma\) is very close to 1, implying classical mechanics are sufficient for calculations.
• As \(v\) approaches the speed of light \,\((c)\), \(\gamma\) becomes significantly larger than 1, indicating the need for relativistic calculations.

In our exercise, \(\gamma\) was found to be approximately 6.125, highlighting the considerable influence of relativistic effects.

Relativistic Speeds

When particles move at relativistic speeds, their velocities are a significant portion of the speed of light.
At these speeds, classical Newtonian physics breaks down, and the effects of relativity become pronounced. Here’s what happens under relativistic conditions:

• Mass increases: The particle's relativistic mass seems to increase as its speed gets closer to light speed, though this is mostly a conceptual aid for understanding momentum at relativistic speeds.
• Time dilation: Time slows down in the moving particle's frame relative to the observer's clock, as detailed previously.
• Length contraction: An object in motion will also appear contracted in its direction of travel from the observer's view.

All these effects come into play with the high-energy particle in the exercise, which moves at 0.992 times the speed of light. Understanding relativistic speeds is essential for applications in modern physics, particularly in particle physics and cosmology.