This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — let's tackle these physics problems. Part 12b(i): Initial accelerations of the particles Step 1: Identify given values and convert units. Given masses: m_1 = 1 kg, m_2 = 2 kg. Separation distance: r = 50 cm = 0.50 m. Gravitational constant: G = 6.67 × 10^-11 N m^2/kg^2. Step 2: Calculate the gravitational force between the particles using Newton's Law of Universal Gravitation. The formula is F = (G m_1 m_2)/(r^2). F = (6.67 × 10^-11 N m^2/kg^2) × (1 kg) × (2 kg)(0.50 m)^2 F = 13.34 × 10^-11 N m^20.25 m^2 F = 53.36 × 10^-11 N F = 5.34 × 10^-10 N Step 3: Calculate the initial acceleration of each particle using Newton's Second Law (F = ma). For the first particle (m_1): a_1 = (F)/(m_1) = 5.34 × 10^-10 N1 kg a_1 = 5.34 × 10^-10 m/s^2 For the second particle (m_2): a_2 = (F)/(m_2) = 5.34 × 10^-10 N2 kg a_2 = 2.67 × 10^-10 m/s^2 The initial accelerations are: a_1 = 5.34 × 10^-10 m/s^2 a_2 = 2.67 × 10^-10 m/s^2 Part 12b(ii): Escape velocity from the moon Step 1: Identify given values and convert units. Mass of the moon: M = 7.4 × 10^22 kg. Radius of the moon: R = 1740 km = 1740 × 10^3 m = 1.74 × 10^6 m. Gravitational constant: G = 6.67 × 10^-11 N m^2/kg^2. Step 2: Calculate the escape velocity using the formula v_e = sqrt((2GM)/(R)). v_e = sqrt(2 × (6.67 × 10^-11) N m^2/kg^2) × (7.4 × 10^22 kg)1.74 × 10^6 m v_e = sqrt(98.616 × 10^11)1.74 × 10^6 m/s v_e = sqrt(56.675 × 10^5) m/s v_e = sqrt(5.6675 × 10^6) m/s v_e ≈ 2380.65 m/s Step 3: Round the result to an appropriate number of significant figures. v_e ≈ 2380 m/s The escape velocity from the moon is: 2380 m/s What's next?