Question

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

Short answer

The proper lifetime of the particle, i.e. the length it would have lasted if it were at rest relative to the detector, is \(4.98 \times 10^{-13} \,s\).

Step-by-step solution

Conversion of Given Values

First convert the length from millimeters to meters: \(1.05 \,mm = 1.05 \times 10^{-3} \,m\). The speed of light (\(c\)) is typically \(3 \times 10^8 \,m/s\).

Apply Basic Kinematics

To determine how long the particle was observed to exist before decaying, use the equation \(v = \frac{d}{t}\). Solve this for \(t\) to get \(t = \frac{d}{v}\). Then, plug in the given values: \(t = \frac{1.05 \times 10^{-3} \, m}{0.992 \times 3 \times 10^8 \, m/s} = 3.53 \times 10^{-12} \,s\). This represents the lifetime that the particle appeared to last before it decayed, as observed by the detector.

Find the Gamma Factor

We need to find the Lorentz Gamma Factor, \(\gamma\), which is calculated using the equation: \(\gamma = \frac{1}{\sqrt{1-(v/c)^2}}\). Substitute the provided values for \(v\) and \(c\): \(\gamma = \frac{1}{\sqrt{1-(0.992)^2}} = 7.089\) (rounded to three decimal places).

Use Time Dilation Equation

Apply the time dilation equation \(t = \gamma t_0\) to solve for \(t_0\), where \(t_0\) is the proper time: \(t_0 = \frac{t}{\gamma} = \frac{3.53 \times 10^{-12}\, s}{7.089}=4.98 \times 10^{-13} \,s\). This is the lifetime the particle would have if it were at rest with respect to the detector.

Key concepts

Time Dilation

Imagine looking at two clocks, one sitting beside you and the other on a spaceship zooming past at nearly the speed of light. If you could see the clock on the spaceship, it would seem to tick more slowly than the one next to you. This phenomenon is known as time dilation, a concept within Einstein's theory of special relativity. Time dilation occurs because the laws of physics are the same for all observers, no matter how fast they are moving. An important consequence is that time passes at different rates for people moving relative to one another.

To observe the effects of time dilation, let's consider the unstable particle from our exercise. Although the particle's track was measured to be 1.05 mm long, the time it took to decay—the lifetime of the particle—appeared longer to the observer because the particle was moving at a speed very close to the speed of light. This effect on time can be confusing, but it's a fundamental aspect of understanding the universe at very high speeds.

Relativistic Kinematics

The motion of objects traveling at significant fractions of the speed of light is described by relativistic kinematics, which corrects classical kinematics to account for the effects of special relativity. Where Newton's laws fall short, relativistic kinematics steps in, allowing us to accurately predict and describe the motion, speed, and energy of objects in high-speed environments. In our textbook's example, to calculate the lifetime of the high-energy particle, we're working with speeds comparable to the speed of light. Consequently, using classical equations without relativistic adjustments wouldn't provide an accurate measure of the particle's lifetime. Using relativistic kinematics, which includes corrections for time dilation, we can determine how phenomena such as high-speed motion affect time and space.

Lorentz Gamma Factor

A key component in the mathematics of special relativity is the Lorentz gamma factor, symbolized as \( \gamma \). It quantifies how much time, length, and mass change for an object moving relative to an observer. The gamma factor is described by the equation \( \gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}} \) where \( v \) is the object's speed and \( c \) is the speed of light in a vacuum. As the speed of an object approaches the speed of light, \( \gamma \) gets larger, indicating significant increases in time dilation and length contraction. For the high-energy particle in our exercise, the Lorentz gamma factor was used to translate the observed lifetime to the proper lifetime—the time it would have actually lasted had it not been moving relative to the detector.

Special Relativity

The theory that revolutionized our understanding of space, time, and energy is special relativity, developed by Albert Einstein in 1905. Special relativity's two postulates state that the laws of physics are the same in all inertial frames of reference, and that the speed of light in a vacuum is the same for all observers, regardless of their relative motion. This theory profoundly impacts relativistic kinematics, leading to effects like time dilation and length contraction, which are noticeable at speeds close to the speed of light. In our exercise, special relativity is indispensable for calculating the proper lifetime of the particle because it provides the framework to understand and apply the time dilation effect.