Question

Three groups of experimenters measure the average decay time for a specific type of radioactive particle. Group 1 accelerates the particles to \(0.5 c,\) moving from left to right, and then measures the decay time in the beam, obtaining a result of 20 ms. Group 2 accelerates the particles to \(-0.5 c,\) from right to left, and then measures the decay time in the beam. Group 3 keeps the particles at rest in a container and measures their decay time. Which of the following is true about these measurements? a) Group 2 measures a decay time of \(20 \mathrm{~ms}\). b) Group 2 measures a decay time less than \(20 \mathrm{~ms}\). c) Group 3 measures a decay time of \(20 \mathrm{~ms}\). d) Both (a) and (c) are true. e) Both (b) and (c) are true.

Short answer

a) Group 2 measures a decay time of 20 ms. b) Group 2 measures a decay time different from 20 ms. c) Group 3 measures a decay time of 20 ms. d) Both (a) and (c) are true. Answer: d) Both (a) and (c) are true.

Step-by-step solution

Calculate the Lorentz factor for Group 1 particles

To calculate the Lorentz factor (also known as the gamma factor), we can use the formula: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) For Group 1 particles moving at \(0.5c\), we have: \(v = 0.5c\) \(\gamma = \frac{1}{\sqrt{1 - \frac{(0.5c)^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.25}} = \frac{1}{\sqrt{0.75}} = \frac{1}{\sqrt{3/4}} = \frac{2}{\sqrt{3}}\)

Calculate the decay time in the rest frame

We can now find the decay time measured in the rest frame (Group 3) using the formula of time dilation: \(\Delta t_0 = \frac{\Delta t}{\gamma}\) where \(\Delta t\) is the decay time measured by Group 1 (20 ms) and \(\Delta t_0\) is the decay time measured by Group 3. Plug in the values: \(\Delta t_0 = \frac{20 \,\text{ms}}{\frac{2}{\sqrt{3}}} = 20 \,\text{ms} \cdot \frac{\sqrt{3}}{2} = 10\sqrt{3} \,\text{ms}\)

Determine the decay time in Group 2 measurements

Since Group 2 is also moving at \(0.5c\), they will also measure a decay time of 20 ms, due to symmetry.

Compare the decay times of Group 2 and Group 3 to the given options

By comparing the decay times of Group 2 and Group 3 with the options given, we can conclude that: a) Group 2 measures a decay time of 20 ms, which is true. c) Group 3 measures a decay time of \(10\sqrt{3} \,\text{ms}\), which is not 20 ms, so this option is false. Hence, the correct option is: d) Both (a) and (c) are true, as Group 2 measures a decay time of 20 ms and Group 3 measures a decay time different from 20 ms.

Key concepts

Lorentz Factor

Understanding the Lorentz factor, often symbolized by \(\gamma\), is crucial when studying motion at speeds comparable to the speed of light. This quantity arises in the theory of special relativity and serves as a scalar factor that determines the amount of time dilation, length contraction, and relativistic mass increase that occurs from the perspective of different inertial reference frames.In radioactive decay experiments where particles move at significant fractions of the speed of light, such as \(0.5c\), we calculate the Lorentz factor using the equation \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(v\) is the velocity of the particle, and \(c\) is the speed of light in a vacuum. The Lorentz factor determines how much time dilates from the perspective of an observer watching the particles move. With \(v = 0.5c\), \(\gamma\) turns into \(\frac{2}{\sqrt{3}}\), indicating that time for these particles is dilated by this factor compared to when they are at rest.

Time Dilation

Time dilation is a fascinating and non-intuitive effect predicted by Einstein's theory of special relativity. It describes how a clock moving relative to an observer ticks slower compared to one that is at rest with respect to that observer. The faster the relative velocity, the greater the effect of time dilation.

Real-World Applications of Time Dilation

• Satellite navigation systems must account for time dilation to maintain accuracy.
• Experiments with fast-moving particles, like the radioactive decay scenario, can only be understood with time dilation in mind.
• Astronauts on the International Space Station experience time slightly slower than those on Earth, though the effect is minimal due to the relatively low orbital velocity.

To determine the true decay time in the rest frame (\(Group 3\)) as seen by stationary observers, we need to take the decay time measured by moving observers (\(Group 1\) in the exercise) and divide it by the Lorentz factor. This calculation allows us to understand how much the decay time was 'stretched out' due to the particles traveling at significant fractions of the speed of light.

Special Relativity

Special relativity, formulated by Albert Einstein in 1905, radically changed our understanding of space, time, and energy. At its core, the theory proposes that the laws of physics are the same for all non-accelerating observers, and, most famously, that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer.

Key Postulates of Special Relativity

• The laws of physics are invariant in all inertial frames of reference.
• The speed of light (\(c\)) is constant in all inertial frames, regardless of the motion of the source or the observer.

The exercise on radioactive particle decay is a practical example of how special relativity applies to high-speed phenomena. Since both Group 1 and Group 2 measure the same decay time of 20 ms when observing the particles moving at \(0.5c\), it is evident how relative motion influences time measurement due to time dilation, a cornerstone result of special relativity. Furthermore, the discrepancy in decay times between Groups 1 and 3 highlights how absolute measurements are not possible without considering the observer's frame of reference.