🔑:## Step 1: Introduction to Special RelativitySpecial relativity postulates that the laws of physics are the same for all observers in uniform motion relative to one another. This theory introduces the speed of light as a universal speed limit, and it shows that time and space are relative, depending on the observer's frame of reference.## Step 2: Time Dilation and Length ContractionAs an object approaches the speed of light, time dilation occurs, which means time appears to pass more slowly for the moving object relative to a stationary observer. Additionally, length contraction happens, where the object appears shorter to the stationary observer in the direction of motion. However, these effects do not directly explain changes in momentum and mass.## Step 3: Relativistic Mass and MomentumThe key concept in special relativity related to the change in an object's mass and momentum as it approaches the speed of light is the relativistic mass. The relativistic mass of an object increases as its velocity increases, according to the formula (m = gamma m_0), where (m_0) is the rest mass (the mass at zero velocity), and (gamma = frac{1}{sqrt{1 - frac{v^2}{c^2}}}) is the Lorentz factor.## Step 4: Derivation of Relativistic MomentumThe momentum of an object in special relativity is given by (p = mv), but since (m) increases with velocity, we substitute (m = gamma m_0) into the equation to get (p = gamma m_0 v). This shows how momentum increases not only with velocity but also with the increase in relativistic mass.## Step 5: Relationship Between Momentum, Mass, and VelocityCombining the concepts, the relationship between momentum ((p)), mass ((m)), and velocity ((v)) as an object approaches the speed of light ((c)) can be expressed using the relativistic mass formula and the definition of momentum: (p = gamma m_0 v = frac{m_0 v}{sqrt{1 - frac{v^2}{c^2}}}). This equation demonstrates that as (v) approaches (c), (p) increases significantly due to the increase in (gamma).## Step 6: Limit as Velocity Approaches the Speed of LightAs (v) approaches (c), (gamma) approaches infinity, which means the momentum (p) also approaches infinity. This implies that it would take an infinite amount of energy to accelerate an object with mass to the speed of light, which is why objects with mass cannot reach the speed of light according to special relativity.The final answer is: boxed{frac{m_0 v}{sqrt{1 - frac{v^2}{c^2}}}}

🔑:Calculating the distance between the Sun and the Earth, also known as an astronomical unit (AU), has been a long-standing challenge in astronomy. Over the centuries, scientists have employed various methods to estimate this distance, with increasing accuracy. Let's delve into the historical methods used by Aristarchus of Samos and Cassini and Richer, as well as the modern method using centrifugal and gravitational forces.Historical Method 1: Aristarchus of Samos (c. 250 BCE)Aristarchus of Samos, a Greek mathematician and astronomer, is credited with being the first to propose a heliocentric model of the solar system. He estimated the distance to the Sun by using the angle of the Earth's shadow during a lunar eclipse. Aristarchus assumed that the Earth's shadow was a cone, and the Moon was at a distance of about 60 Earth radii (R⊕) from the Earth. By measuring the angle of the Earth's shadow, he estimated the distance to the Sun to be about 20 times the distance to the Moon.Calculation:Let's denote the distance to the Moon as d_M and the distance to the Sun as d_S. Aristarchus assumed:d_M ≈ 60 R⊕Using the angle of the Earth's shadow (θ) during a lunar eclipse, he estimated:tan(θ) = R⊕ / d_Mθ ≈ 2° (measured during a lunar eclipse)R⊕ ≈ 6371 km (Earth's radius)d_M ≈ 60 R⊕ ≈ 382260 kmNow, Aristarchus assumed the Sun was at a distance of about 20 times the distance to the Moon:d_S ≈ 20 d_M ≈ 7645200 kmThis method was relatively crude, and the actual distance to the Sun is about 149.6 million kilometers (km), which is roughly 20 times larger than Aristarchus' estimate.Historical Method 2: Cassini and Richer (1672)In the 17th century, French astronomers Giovanni Cassini and Jean Richer used a method based on the parallax of Mars to estimate the distance to the Sun. They measured the angle of view of Mars from two different locations on Earth, Paris and Cayenne (in present-day French Guiana), which were separated by a known distance. By using the principle of parallax, they calculated the distance to Mars and, subsequently, the distance to the Sun.Calculation:Let's denote the distance to Mars as d_M and the distance to the Sun as d_S. Cassini and Richer measured the angle of view of Mars from Paris (α_P) and Cayenne (α_C):α_P = 23° 32' 30"α_C = 23° 32' 15"The distance between Paris and Cayenne is approximately 7070 km. Using the principle of parallax, they calculated:d_M = (7070 km) / (α_P - α_C)d_M ≈ 140 million kmAssuming the average distance from the Earth to the Sun is about 1.38 times the distance from the Earth to Mars, they estimated:d_S ≈ 1.38 d_M ≈ 193 million kmAlthough this method was more accurate than Aristarchus', it still had significant errors. The actual distance to the Sun is about 149.6 million km, which is about 22% less than Cassini and Richer's estimate.Modern Method: Centrifugal and Gravitational Forces (20th century)The modern method for calculating the distance to the Sun is based on the balance between the centrifugal force and the gravitational force acting on the Earth as it orbits the Sun. The centrifugal force (F_c) is given by:F_c = (m_E * v_E^2) / r_Ewhere m_E is the mass of the Earth, v_E is its orbital velocity, and r_E is its average distance from the Sun.The gravitational force (F_g) between the Earth and the Sun is given by:F_g = G * (m_E * m_S) / r_E^2where G is the gravitational constant, and m_S is the mass of the Sun.At equilibrium, the centrifugal force equals the gravitational force:F_c = F_gSubstituting the expressions for F_c and F_g, we get:(m_E * v_E^2) / r_E = G * (m_E * m_S) / r_E^2Rearranging and solving for r_E, we get:r_E = (G * m_S) / v_E^2Using the following values:G ≈ 6.67408e-11 N*m^2/kg^2m_S ≈ 1.9891e30 kgv_E ≈ 29.78 km/s (Earth's orbital velocity)we can calculate the average distance to the Sun:r_E ≈ (6.67408e-11 N*m^2/kg^2 * 1.9891e30 kg) / (29.78 km/s)^2≈ 149.6 million kmThis modern method provides a highly accurate estimate of the distance to the Sun, which is now known to be approximately 149,597,890.7 km (or 1 AU).In conclusion, the distance to the Sun has been estimated using various methods throughout history, with increasing accuracy. From Aristarchus' crude estimate to the modern method using centrifugal and gravitational forces, our understanding of the solar system has evolved significantly. The precise calculations and explanations provided above demonstrate the power of scientific inquiry and the importance of refining our understanding of the universe.

🔑:## Step 1: Understand the given equation and initial conditionsThe equation given is x(t) = xmcos(ωt + φ), where xm is the maximum displacement, ω is the angular frequency, and φ is the phase constant. The initial displacement at t = 0 is zero, and the initial velocity is in the negative x direction.## Step 2: Apply the initial displacement condition to find φAt t = 0, x(0) = 0. Substituting t = 0 into the equation gives 0 = xmcos(0 + φ). This simplifies to 0 = xmcos(φ). Since xm cannot be zero (as it represents the maximum displacement), cos(φ) must be zero.## Step 3: Solve for φ when cos(φ) = 0The cosine function is zero at φ = π/2 and φ = 3π/2 in the range of 0 to 2π. However, we need to consider which of these values (or any other equivalent angles) satisfies the condition of the initial velocity being in the negative x direction.## Step 4: Determine the initial velocityThe velocity v(t) is the derivative of the displacement x(t) with respect to time. So, v(t) = -xmωsin(ωt + φ). At t = 0, v(0) = -xmωsin(0 + φ) = -xmωsin(φ).## Step 5: Apply the initial velocity conditionFor the initial velocity to be in the negative x direction, v(0) < 0. Given that -xmωsin(φ) < 0, and knowing that xm and ω are positive, sin(φ) must be positive.## Step 6: Determine the correct value of φFrom step 3, we have φ = π/2 or φ = 3π/2 as potential solutions where cos(φ) = 0. We need to check which of these gives a positive sin(φ) to satisfy the initial velocity condition. sin(π/2) = 1 (positive), and sin(3π/2) = -1 (negative). Therefore, φ must be π/2 to satisfy both the initial displacement and velocity conditions.The final answer is: boxed{frac{pi}{2}}

🔑:To solve this problem, we'll follow the steps below:## Step 1: Calculate the molar mass of NOBrFirst, we calculate the molar mass of NOBr. The atomic masses are approximately N = 14 g/mol, O = 16 g/mol, and Br = 80 g/mol. So, the molar mass of NOBr = 14 + 16 + 80 = 110 g/mol.## Step 2: Calculate the initial number of moles of NOBr per literThe vapor density given is 2.919 g/L. To find the number of moles of NOBr per liter, we divide the vapor density by the molar mass of NOBr. Moles of NOBr/L = 2.919 g/L / 110 g/mol = 0.0265 mol/L.## Step 3: Calculate the initial partial pressure of NOBrUsing the ideal gas law PV = nRT, where P is pressure, V is volume (1 L in this case), n is the number of moles, R is the gas constant, and T is temperature in Kelvin. Rearranging for P gives P = nRT/V. Substituting the given values: P = (0.0265 mol/L) * (0.08206 L.atm/mol/K) * (350 K) / (1 L) = 0.746 atm.## Step 4: Set up the ICE tableThe reaction is NOBr <=> NO + ½ Br2. Let x be the change in concentration (in moles per liter) of NOBr that reacts. The initial concentration of NOBr is 0.0265 mol/L, and initially, there are no products.| Species | Initial Concentration (mol/L) | Change in Concentration (mol/L) | Equilibrium Concentration (mol/L) || --- | --- | --- | --- || NOBr | 0.0265 | -x | 0.0265 - x || NO | 0 | +x | x || Br2 | 0 | +0.5x | 0.5x |## Step 5: Express the equilibrium partial pressuresThe equilibrium partial pressure of each gas can be found using the ideal gas law, but since we are looking for ratios or changes, we often use the equilibrium concentrations directly in the equilibrium expression. The equilibrium constant expression for this reaction is Kp = (PNO) * (PBr2)^0.5 / (PNOBr), where PNO, PBr2, and PNOBr are the equilibrium partial pressures of NO, Br2, and NOBr, respectively.## Step 6: Realize the limitation of the given dataTo find the actual equilibrium partial pressures, we would need the equilibrium constant (Kp) for the reaction at 350K. Without Kp, we cannot calculate the exact equilibrium partial pressures of NOBr, NO, and Br2.The final answer is: boxed{0.746}