This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Identify the given values and constants. Mass of Earth (m_E) = 5.98 × 10^24 kg Mass of Sun (M_S) = 1.99 × 10^30 kg Distance between Earth and Sun (r) = 160 million km Gravitational constant (G) = 6.674 × 10^-11 N · m^2/kg^2 Step 2: Convert the distance from kilometers to meters. 1 million km = 10^6 km 160 million km = 160 × 10^6 km = 1.6 × 10^8 km Since 1 km = 1000 m = 10^3 m: r = 1.6 × 10^8 km × 10^3 m1 km = 1.6 × 10^11 m Step 3: Use the formula for gravitational potential energy. The gravitational potential energy U between two masses M and m separated by a distance r is given by: U = -(GMm)/(r) Substitute the values for the Sun and Earth: U = -(6.674 × 10^-11 N · m^2/kg^2)(1.99 × 10^30 kg)(5.98 × 10^24 kg)1.6 × 10^11 m Step 4: Calculate the numerator. G M_S m_E = (6.674 × 10^-11) × (1.99 × 10^30) × (5.98 × 10^24) G M_S m_E = (6.674 × 1.99 × 5.98) × (10^-11 × 10^30 × 10^24) G M_S m_E ≈ 79.40 × 10^(-11+30+24) G M_S m_E ≈ 79.40 × 10^43 N · m^2 Step 5: Calculate the gravitational potential energy. U = -79.40 × 10^43 N · m^21.6 × 10^11 m U = -((79.40)/(1.6)) × (10^4310^11) J U = -49.625 × 10^(43-11) J U = -49.625 × 10^32 J U = -4.9625 × 10^33 J The gravitational potential energy of the Earth is: -4.96 × 10^33 J