🔑:## Step 1: Convert the given values to SI units for consistencyThe charge q1 = 5μC = 5 * 10^-6 C, and the charge on the small sphere q2 = 2μC = 2 * 10^-6 C. The mass of the sphere m = 4g = 0.004 kg. The initial speed v1 = 40.0 m/s, and the speed at the point of interest v2 = 25.0 m/s. The initial distance r1 = 6.00 cm = 0.06 m.## Step 2: Determine the initial potential energy using Coulomb's lawThe initial potential energy (U1) can be calculated using the formula U = k * q1 * q2 / r, where k is Coulomb's constant (approximately 8.99 * 10^9 N*m^2/C^2). Thus, U1 = k * q1 * q2 / r1.## Step 3: Calculate the initial potential energyU1 = (8.99 * 10^9 N*m^2/C^2) * (5 * 10^-6 C) * (2 * 10^-6 C) / (0.06 m) = (8.99 * 10^9) * (5 * 2) * (10^-6 * 10^-6) / 0.06 = (8.99 * 10^9) * 10 * 10^-12 / 0.06 = 89.9 * 10^-3 / 0.06 = 1.4985 J.## Step 4: Determine the initial kinetic energyThe initial kinetic energy (K1) is given by the formula K = 0.5 * m * v^2. Thus, K1 = 0.5 * m * v1^2.## Step 5: Calculate the initial kinetic energyK1 = 0.5 * (0.004 kg) * (40.0 m/s)^2 = 0.5 * 0.004 * 1600 = 3.2 J.## Step 6: Determine the total initial energyThe total initial energy (E1) is the sum of the initial potential energy (U1) and the initial kinetic energy (K1). E1 = U1 + K1.## Step 7: Calculate the total initial energyE1 = 1.4985 J + 3.2 J = 4.6985 J.## Step 8: Determine the final potential energy when the sphere's speed is 25.0 m/sTo find the final potential energy (U2), we need the distance (r2) at the point when the speed is 25.0 m/s. However, since we are using conservation of energy, we can relate the initial and final energies without explicitly calculating r2.## Step 9: Determine the final kinetic energyThe final kinetic energy (K2) is given by K2 = 0.5 * m * v2^2.## Step 10: Calculate the final kinetic energyK2 = 0.5 * (0.004 kg) * (25.0 m/s)^2 = 0.5 * 0.004 * 625 = 1.25 J.## Step 11: Apply conservation of energy to find the final potential energyThe total final energy (E2) equals the total initial energy (E1), so E2 = E1. Since E2 = U2 + K2, we can find U2 by rearranging: U2 = E1 - K2.## Step 12: Calculate the final potential energyU2 = 4.6985 J - 1.25 J = 3.4485 J.## Step 13: Use Coulomb's law to find the distance at the final pointRearrange the formula for potential energy to solve for r2: r2 = k * q1 * q2 / U2.## Step 14: Calculate the distance at the final pointr2 = (8.99 * 10^9 N*m^2/C^2) * (5 * 10^-6 C) * (2 * 10^-6 C) / (3.4485 J) = (8.99 * 10^9) * 10 * 10^-12 / 3.4485 = 89.9 * 10^-3 / 3.4485 = 0.02607 m.## Step 15: Calculate the acceleration using Coulomb's lawThe force (F) on the sphere is given by F = k * q1 * q2 / r^2. The acceleration (a) is found by F = m * a, so a = F / m.## Step 16: Calculate the force at the final pointF = (8.99 * 10^9 N*m^2/C^2) * (5 * 10^-6 C) * (2 * 10^-6 C) / (0.02607 m)^2.## Step 17: Perform the calculation for forceF = (8.99 * 10^9) * 10 * 10^-12 / (0.02607)^2 = (8.99 * 10^9) * 10 * 10^-12 / (0.0006799) = 89.9 * 10^-3 / 0.0006799 = 132.07 N.## Step 18: Calculate the accelerationa = F / m = 132.07 N / 0.004 kg.## Step 19: Perform the final calculation for accelerationa = 132.07 / 0.004 = 33017.5 m/s^2.The final answer is: boxed{33017.5}

🔑:To determine whether an 'unphysical' solution to an equation that is invariant with respect to time-translation and time-reversal is actually unphysical or a valid physical solution corresponding to alternate boundary conditions, we can follow a systematic approach.## Step 1: Understand the SymmetriesFirst, we need to understand the implications of the equation being invariant under time-translation and time-reversal. Time-translation invariance means that the laws of physics do not change over time, implying that if a certain situation occurs, the same situation will occur again after a certain time interval. Time-reversal invariance means that the laws of physics are the same if time runs backwards, which implies that for every physical process, there is a corresponding process where time is reversed.## Step 2: Analyze the EquationNext, analyze the equation describing the physical system. If the equation is invariant under time-translation and time-reversal, any solution to the equation should also exhibit these symmetries. However, some solutions might seem 'unphysical' at first glance because they do not match our intuitive understanding of the physical world.## Step 3: Consider Boundary ConditionsConsider the boundary conditions applied to the system. Different boundary conditions can lead to different solutions of the same equation. An 'unphysical' solution might become physical if we consider alternate boundary conditions that are consistent with the symmetries of the equation.## Step 4: General ExpressionA general expression of this concept can be formulated as follows: Given a physical system described by an equation (E) that is invariant under time-translation (T) and time-reversal (R), a solution (S) that seems unphysical under standard boundary conditions (B) should be reevaluated under alternate boundary conditions (B') that respect the (T) and (R) symmetries. The solution (S) can be considered physical if there exists a (B') for which (S) satisfies (E) and the symmetries (T) and (R).## Step 5: ImplicationsThe implications of this concept are significant. It suggests that our understanding of physical systems should be broadened to include solutions that might initially seem unphysical but could be valid under different boundary conditions. This challenges our intuitive understanding and encourages a more mathematical and systematic approach to understanding physical phenomena.The final answer is: boxed{S}

🔑:## Step 1: Understanding Quarks and Their ColorsQuarks are elementary particles that come in six flavors: up (u), down (d), charm (c), strange (s), top (t), and bottom (b). Each of these flavors can have one of three colors: red, green, or blue. This color charge is the basis for the strong nuclear force, which holds quarks together inside protons and neutrons, and these in turn inside atomic nuclei.## Step 2: Formation of MesonsMesons are particles formed by a quark and an antiquark. For a meson to be colorless, it must combine a quark of one color with an antiquark of the corresponding anticolor (e.g., a red quark with an antired antiquark). This combination ensures that the meson has no net color charge, making it colorless and thus stable against the strong nuclear force. Examples of mesons include the pion (π, composed of u and d quarks or their antiquarks) and the kaon (K, composed of s and u or d quarks and their antiquarks).## Step 3: Formation of BaryonsBaryons are particles composed of three quarks. To be colorless, a baryon must contain one quark of each color: red, green, and blue. This combination ensures that the baryon has no net color charge, making it stable against the strong nuclear force. The most common baryons are the proton (composed of two u quarks and one d quark) and the neutron (composed of two d quarks and one u quark). Other examples include the lambda (Λ, composed of u, d, and s quarks) and the sigma (Σ, composed of u, d, and s quarks in different combinations).## Step 4: Focus on u, d, s QuarksThe u, d, and s quarks are often focused on when discussing common baryons and mesons because they are the lightest quarks and thus the most stable and abundant in nature. The charm, top, and bottom quarks are heavier and decay quickly into the lighter quarks, making particles composed of these quarks less common and typically observed in high-energy particle physics experiments. The u and d quarks are particularly important as they form the protons and neutrons that make up atomic nuclei, while the s quark is involved in the formation of particles like kaons and hyperons, which are important in understanding the strong nuclear force and the structure of matter.The final answer is: There is no specific numerical answer to this problem as it involves explanatory text about the formation of mesons and baryons from quarks and the reasons for focusing on certain quark flavors.

🔑:The primary reason why Jupiter is composed mostly of gas is due to its unique combination of gravitational pull, pressure, and temperature conditions. These factors work together to create an environment that is conducive to a gaseous state, rather than a solid or liquid one.Gravitational Pull:Jupiter's massive size and gravitational pull play a crucial role in its gaseous composition. With a mass approximately 318 times that of Earth, Jupiter's gravity is incredibly strong, pulling in and holding onto the surrounding material in the solar nebula during its formation. This strong gravity led to the accumulation of a massive amount of gas, primarily hydrogen and helium, which were the most abundant elements in the solar nebula. The gravity was so strong that it prevented the gas from escaping, allowing it to collapse and form a massive gas giant.Pressure:The pressure at Jupiter's core is estimated to be around 200-250 GPa (gigapascals), which is about 2-2.5 million times the pressure at Earth's core. This extreme pressure is a result of the planet's massive size and gravity. However, despite this intense pressure, the temperature at Jupiter's core is still relatively low, around 20,000 Kelvin (35,000°F). At these conditions, the gas is unable to condense into a liquid or solid, and instead remains in a gaseous state.Temperature:The temperature at Jupiter's core is relatively low compared to other planets, which is another factor contributing to its gaseous composition. The core temperature is not high enough to ionize the gas, which would allow it to conduct electricity and potentially lead to the formation of a solid or liquid state. Instead, the gas remains in a state of high-pressure, high-temperature gas, often referred to as a "degenerate gas" or "metallic hydrogen."Additional Factors:Other factors also contribute to Jupiter's gaseous composition, including:1. Lack of a solid surface: Jupiter's atmosphere gradually transitions into a liquid hydrogen and helium layer, and then into a dense, metallic hydrogen core. There is no clear boundary between the atmosphere and the core, which means that there is no solid surface to speak of.2. Convection and mixing: Jupiter's atmosphere is dynamic, with strong convection currents and mixing of gases. This helps to distribute heat and maintain a relatively uniform temperature throughout the planet, preventing the formation of solid or liquid layers.3. Chemical composition: Jupiter's atmosphere is primarily composed of hydrogen and helium, which are both gases at the temperatures and pressures found on the planet. The presence of these gases, combined with the lack of heavier elements, contributes to the planet's gaseous state.In summary, the primary reason why Jupiter is composed mostly of gas is due to the combination of its strong gravitational pull, extreme pressure, and relatively low temperature conditions. These factors work together to create an environment that is conducive to a gaseous state, preventing the formation of a solid or liquid planet.