Question

The mean diameters of planets A and B are 9.3 × 103 km and 1.6 × 104 km, respectively. The ratio of the mass of planet A to that of planet B is 0.76. (a) What is the ratio of the mean density of A to that of B? (b) What is the ratio of the gravitational acceleration on A to that on B? (c) What is the ratio of escape speed on A to that on B?

Answer 1

volume of the planet = (4/3) * pi * (diameter / 2)^3

density = mass / volume

let the mass of planet B = m so

mass of planet A = 0.76 * m

density of planet A = 0.76 m / ((4/3) * pi * (9.3 * 10^6 / 2)^3)

density of planet B = m / ((4/3) * pi * (1.6 * 10^7 / 2)^3)

ratio of mean density = 0.76 m / ((4/3) * pi * (9.3 * 10^6 / 2)^3) / m / ((4/3) * pi * (1.6 * 10^7 / 2)^3)

ratio of mean density = 3.87

acceleration due to gravity = G * mass of planet / radius of planet^2

acceleration due to gravity of planet A = G * 0.76 * m / (9.3 * 10^6)^2

acceleration due to gravity of planet B = G * m / (1.6 * 10^7)^2

ratio of gravitation acceleration = G * 0.76 * m / (9.3 * 10^6)^2 / G * m / (1.6 * 10^7)^2

ratio of gravitation acceleration = 2.2495

escape velocity = sqrt(2 * G * mass / radius)

escape velocity of planet A = sqrt(2 * G * 0.76 * m / (9.3 * 10^6))

escape velocity of planet B = sqrt(2 * G * m / (1.6 * 10^7))

ratio of escape velocity = sqrt(2 * G * 0.76 * m / (9.3 * 10^6)) / sqrt(2 * G * m / (1.6 * 10^7))

ratio of escape velocity = sqrt(0.76 / (9.3 * 10^6)) / sqrt(1 / (1.6 * 10^7))

ratio of escape velocity = 1.1434