🔑:## Step 1: Understand the given problem and the equation of special relativity.The problem involves understanding the concept of time dilation in special relativity, where the time measured by an observer in motion (Donald Duck) is different from the time measured by a stationary observer (Ludwig von Drake). The equation given is t = t0 / sqrt(1 - v^2/c^2), where t is the time measured by Ludwig, t0 is the time measured by Donald, v is the speed of the bomb, and c is the speed of light.## Step 2: Identify the relationship between the half-lives measured by Ludwig and Donald.It's given that Donald's measurement of the half-life is a factor of 2 smaller than Ludwig's. This means t0 = t/2.## Step 3: Substitute the relationship between t0 and t into the special relativity equation.Substitute t0 = t/2 into the equation t = t0 / sqrt(1 - v^2/c^2) to find the relationship between t and v.## Step 4: Solve for v in terms of c.By substituting t0 = t/2 into the equation, we get t = (t/2) / sqrt(1 - v^2/c^2). This simplifies to 2 = 1 / sqrt(1 - v^2/c^2). Squaring both sides gives 4 = 1 / (1 - v^2/c^2), which further simplifies to 1 - v^2/c^2 = 1/4. Solving for v^2/c^2 gives v^2/c^2 = 3/4. Taking the square root of both sides, we find v/c = sqrt(3)/2.The final answer is: boxed{frac{sqrt{3}}{2}}

🔑:A telescope with an f/4.6 focal ratio has a relatively fast optical system, which means it has a shorter focal length compared to its primary mirror or lens diameter. This focal ratio has significant implications on the telescope's design, performance, and user experience.Implications on Telescope Design:1. Optical Design: A fast focal ratio like f/4.6 requires a more complex optical design to correct for aberrations, such as spherical aberration, coma, and astigmatism. This may involve the use of aspheric mirrors, corrective lenses, or advanced optical coatings.2. Mirror or Lens Size: To achieve a given focal length, a faster focal ratio requires a larger primary mirror or lens diameter. This can increase the telescope's size, weight, and cost.3. Tube Length and Stability: A shorter focal length results in a shorter tube length, which can make the telescope more compact and portable. However, this can also lead to stability issues, as the telescope may be more prone to vibrations and flexure.Impact on Astrophotography:1. Field of View: A fast focal ratio provides a wider field of view, making it ideal for capturing large celestial objects, such as nebulae, star clusters, or galaxies.2. Exposure Time: The faster focal ratio allows for shorter exposure times, which is beneficial for imaging faint objects or capturing high-resolution images of the Moon and planets.3. Image Scale: However, the faster focal ratio may result in a smaller image scale, which can make it more challenging to capture high-resolution images of smaller objects, such as double stars or planetary details.Impact on Visual Observation:1. Field of View: The wider field of view provided by the f/4.6 focal ratio is beneficial for visual observation, allowing observers to take in more of the celestial landscape at once.2. Eyepiece Selection: The faster focal ratio may require the use of shorter-focal-length eyepieces to achieve a comfortable exit pupil and magnification. This can be a challenge, as shorter eyepieces may not be as comfortable to use or may not provide the same level of eye relief.3. Image Brightness: The faster focal ratio can result in a brighter image, making it easier to observe faint objects. However, this may also increase the impact of light pollution and sky glow.Manufacturing Challenges:1. Optical Fabrication: The faster focal ratio requires more precise optical fabrication, which can increase the manufacturing cost and complexity.2. Corrective Optics: The use of corrective optics, such as aspheric mirrors or lenses, can add to the manufacturing cost and challenge.3. Quality Control: Ensuring the optical quality and accuracy of the telescope's components is crucial, as small errors can have a significant impact on the telescope's performance.Trade-offs between Telescope Designs:1. Focal Ratio vs. Aperture: A faster focal ratio often requires a larger aperture to maintain a given level of image quality, which can increase the telescope's size, weight, and cost.2. Optical Quality vs. Cost: The use of advanced optical designs and materials can improve the telescope's performance but may increase the cost.3. Portability vs. Stability: A more compact telescope design may be more portable but may also be more prone to stability issues, which can affect image quality.Influence on Eyepiece Selection:1. Focal Length: The choice of eyepiece focal length will depend on the desired magnification and exit pupil. Shorter eyepieces may be required to achieve a comfortable magnification with the f/4.6 focal ratio.2. Eye Relief: The eyepiece's eye relief becomes more critical with faster focal ratios, as the exit pupil may be smaller. Eyepieces with longer eye relief can provide a more comfortable viewing experience.3. Field of View: The eyepiece's field of view should be matched to the telescope's field of view to minimize vignetting and ensure a comfortable viewing experience.Overall User Experience:1. Image Quality: The f/4.6 focal ratio can provide excellent image quality, with a wide field of view and good contrast.2. Ease of Use: The telescope's design and eyepiece selection can impact the overall user experience. A well-designed telescope with comfortable eyepieces can make observing more enjoyable and rewarding.3. Versatility: The f/4.6 focal ratio can be versatile, suitable for a range of observing applications, from deep-sky astrophotography to planetary observation.In conclusion, the f/4.6 focal ratio has significant implications on the telescope's design, performance, and user experience. While it offers advantages in terms of field of view, image brightness, and exposure time, it also presents challenges in terms of optical design, manufacturing, and eyepiece selection. The choice of focal ratio is a critical consideration in telescope design, and the f/4.6 focal ratio can be an excellent choice for observers who value a wide field of view, good image quality, and versatility.

🔑:## Step 1: Calculate the total volume of the ammonia in the tank.First, we need to calculate the total volume of the ammonia. The total mass of ammonia is given as 2 kg, and the quality is 50%. The specific volume of the saturated liquid (Vf) is 0.001638 m^3/kg, and the specific volume of the saturated vapor (Vg) is 0.1422 m^3/kg. The total specific volume (V) can be calculated using the formula V = Vf + x(Vg - Vf), where x is the quality.## Step 2: Apply the formula to calculate the total specific volume.Substitute the given values into the formula: V = 0.001638 + 0.5(0.1422 - 0.001638) = 0.001638 + 0.5(0.140562) = 0.001638 + 0.070281 = 0.071919 m^3/kg.## Step 3: Calculate the total volume of the ammonia.The total volume (V_total) of the ammonia can be found by multiplying the total mass by the total specific volume: V_total = 2 kg * 0.071919 m^3/kg = 0.143838 m^3.## Step 4: Calculate the volume of the liquid ammonia.The volume of the liquid ammonia (V_liquid) can be calculated by multiplying the mass of the liquid ammonia by the specific volume of the saturated liquid. Since the quality is 50%, the mass of the liquid ammonia is 50% of the total mass, which is 1 kg. Therefore, V_liquid = 1 kg * 0.001638 m^3/kg = 0.001638 m^3.## Step 5: Calculate the mass of liquid that can be removed.The mass of liquid that can be removed through the valve can be calculated by considering that the temperature and pressure remain constant, thus the quality and the specific volumes of the liquid and vapor phases remain the same. Since the valve is at the bottom, only liquid can be removed until the tank contains only saturated vapor at the given temperature and quality. However, the actual limiting factor is the amount of liquid present initially. Given that we have 1 kg of liquid (from the 50% quality of the 2 kg total), and we want to know how much of this liquid can be removed, we must consider the process in terms of the system's overall behavior.## Step 6: Consider the process of removing liquid.As liquid is removed, the total mass decreases, but the quality (which is the mass fraction of vapor) increases because more of the remaining mass is vapor. However, since the temperature is constant, and we are removing only liquid, the vapor mass remains constant until all liquid is removed. The vapor mass can be calculated as 50% of the initial total mass (since the quality is 50%), which is 1 kg. This means that as we remove liquid, we are essentially reducing the total mass while keeping the vapor mass constant at 1 kg until no liquid remains.## Step 7: Determine the maximum amount of liquid that can be removed.Given that we start with 1 kg of liquid and the vapor mass is constant at 1 kg, we can remove liquid until only vapor remains. Thus, the maximum amount of liquid that can be removed is equal to the initial mass of liquid, which is 1 kg.The final answer is: boxed{1}

🔑:In Quantum Field Theory (QFT), the creation of real particles from the vacuum is a fascinating process that involves the interplay of virtual particles, quantum fluctuations, and the energy-momentum uncertainty principle. Pair production, where a particle-antiparticle pair is created from the vacuum, is a fundamental process that illustrates the concept of particle creation in QFT.Virtual particles and the vacuumIn QFT, the vacuum is not a completely empty state, but rather a dynamic medium that is filled with virtual particles and antiparticles. These virtual particles are "off-shell," meaning they do not satisfy the usual energy-momentum relation, E² = (pc)^2 + (mc^2)^2, where E is the energy, p is the momentum, m is the mass, and c is the speed of light. Virtual particles are constantly appearing and disappearing in the vacuum, with their existence limited by the Heisenberg uncertainty principle, ΔE Δt ≥ ħ/2, where ΔE is the energy uncertainty, Δt is the time uncertainty, and ħ is the reduced Planck constant.Pair creation from vacuumThe process of pair creation from vacuum involves the conversion of energy from an external source, such as a strong electromagnetic field or a high-energy particle collision, into a particle-antiparticle pair. This process is often referred to as "pair production." The energy required to create a pair of particles is at least twice the rest mass energy of the particles, 2mc^2, where m is the mass of the particle.The process of pair creation can be understood as follows:1. Virtual particle-antiparticle pair creation: A virtual particle-antiparticle pair is created from the vacuum, with the particle and antiparticle having opposite charges and spins.2. Energy transfer: The external energy source, such as an electromagnetic field, transfers energy to the virtual particle-antiparticle pair, promoting them to "on-shell" status, where they satisfy the energy-momentum relation.3. Pair creation: The energy transferred to the virtual pair is sufficient to create a real particle-antiparticle pair, which is then emitted from the vacuum.Role of virtual particlesVirtual particles play a crucial role in the process of pair creation. They provide a "seed" for the creation of real particles, allowing the energy from the external source to be transferred to the particle-antiparticle pair. Virtual particles also facilitate the creation of particles with the correct quantum numbers, such as charge, spin, and momentum.Implications of pair creationThe process of pair creation from vacuum has significant implications for our understanding of particle physics:1. Vacuum structure: The existence of virtual particles and the process of pair creation demonstrate that the vacuum is a dynamic, non-empty state, which is a fundamental aspect of QFT.2. Particle creation: Pair creation illustrates the concept of particle creation from the vacuum, which is a key feature of QFT and has been experimentally verified in various contexts, such as high-energy particle collisions and astrophysical phenomena.3. Quantum fluctuations: The process of pair creation is a manifestation of quantum fluctuations, which are inherent to QFT and play a crucial role in many phenomena, including particle creation, decay, and scattering.4. Cosmological implications: The process of pair creation from vacuum has implications for our understanding of the early universe, where particle creation and annihilation played a crucial role in shaping the universe's evolution.Experimental evidencePair creation from vacuum has been experimentally verified in various contexts, including:1. High-energy particle collisions: Particle colliders, such as the Large Hadron Collider (LHC), have created particle-antiparticle pairs from the vacuum, demonstrating the process of pair creation.2. Astrophysical phenomena: Pair creation has been observed in astrophysical contexts, such as in the vicinity of black holes, neutron stars, and during gamma-ray bursts.3. Quantum optics: Pair creation has been demonstrated in quantum optics experiments, where photons are created from the vacuum through the process of spontaneous parametric down-conversion.In conclusion, the creation of real particles from the vacuum in Quantum Field Theory, specifically through pair production, is a fundamental process that involves the interplay of virtual particles, quantum fluctuations, and the energy-momentum uncertainty principle. The process of pair creation has significant implications for our understanding of particle physics, vacuum structure, and cosmology, and has been experimentally verified in various contexts.