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 of the particle is approximately \(2.80 \times 10^{-11} \text{ s}\).

Step 1: Gather Known Variables

We know the following variables from the problem:- Track length, \( L = 1.05 \text{ mm} \)- Speed of the particle, \( v = 0.992c \)- Speed of light, \( c \approx 3 \times 10^{8} \text{ m/s} \)

Step 2: Calculate Time in Detector Frame

In the frame of the detector, the length the particle travels can be related to the time it took to decay using \( L = v \cdot t' \).Rearranging gives the time \( t' \) as: \[ t' = \frac{L}{v} = \frac{1.05 \times 10^{-3} \text{ m}}{0.992 \times 3 \times 10^{8} \text{ m/s}} \approx 3.54 \times 10^{-12} \text{ s} \]

Step 3: Use Time Dilation Formula

The time dilation formula relates the proper time \( t \) to the observed time in the detector frame \( t' \) by:\[ t = \frac{t'}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]

Step 4: Solve for Proper Lifetime

Substitute \( t' = 3.54 \times 10^{-12} \text{ s} \) and \( v = 0.992c \) into the time dilation formula:\[ t = \frac{3.54 \times 10^{-12}}{\sqrt{1 - (0.992)^2}} \]Calculate the denominator: \[ \sqrt{1 - 0.992^2} = \sqrt{1 - 0.984064} = \sqrt{0.015936} \approx 0.1263 \]Now, solve for \( t \):\[ t \approx \frac{3.54 \times 10^{-12}}{0.1263} \approx 2.80 \times 10^{-11} \text{ s} \]

Key Concepts

Understanding Time Dilation

Time dilation is a fascinating concept from the theory of relativity that describes how time can stretch or shorten depending on the speed of an object relative to a stationary observer.
This concept means that time is not universal and can vary based on velocity and gravitational fields. In our example, the particle is moving at a significant fraction of the speed of light, specifically at 0.992c.
As such, its "proper time" - the time experienced by the particle - actually passes more slowly compared to the time in the detector's frame. This means that while the particle moves fast, its internal clock ticks slower when compared to an observer at rest in the detector frame. The time dilation formula, \[ t = \frac{t'}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]helps us calculate the proper lifetime of the particle by correcting the observer's view (detector frame) to find out how long the particle would exist if observed at rest relative to the detector.The implication of time dilation is profound; it tells us that at very high speeds, neat everyday notions of time need adjustment, bending to fit the principles of special relativity.

The Process of Particle Decay

Particles, especially those that aren’t stable, can decay into other particles, during this process, the original particle ceases to exist. This is a common phenomenon in physics and is especially noted among subatomic particles. Particle decay is generally described through its "lifetime" which tells us how long, on average, a particle exists before it decays.
For the high-energy particle in this problem, we are interested in knowing its proper lifetime. At rest with respect to the detector, this proper lifetime represents how long the particle would exist before decaying, without the dilation effects due to its high velocity.The combination of particle decay and relativity messages that while a short track is left in a detector due to decay (here, 1.05 mm), it doesn’t necessarily mean the particle decayed quickly from its own perspective (proper time).
From the frame of the detector, calculations considering its speed and path show us an elongated experience of time. This is because the high velocity causes time for the particle to slow down—illustrated by the dilation effect, leading to a calculated proper lifetime of approximately \[ 2.80 \times 10^{-11} \text{ s} \].

Exploring Relativity

Relativity, proposed by Albert Einstein, revolutionized how we understand space and time. It has two primary forms: General Relativity and Special Relativity, with our focus here being the latter. Special Relativity primarily considers observers in inertial frames of motion and dictates that the laws of physics are the same for these observers.
A core aspect of Special Relativity is how it deals with high speeds, particularly those close to the speed of light. Two important principles include:

• Time Dilation: Mentioned earlier, where time moves slower for fast-moving objects relative to a stationary observer.
• Length Contraction: Where objects are measured to be shorter in the direction they are traveling, from an observer's stationary perspective.

Combining these, relativity mandates that as fast-moving particles traverse space, like our high-energy particle, their experience - referred to as "proper lifetime" - differs from that of a stationary observer. Our understanding of particle decay and their interaction with relativity doesn’t just describe these phenomena, but links them together, resulting in a coherent analysis of what truly transpires at subatomic levels. This blend of concepts shows the interconnected world of particles and the exotic rules of high-speed physics.