Question

The mean diameters of X and Y, two planets in the same solar system, are 6.4 ✕ 10³ km and 2.0 ✕ 10⁴ km, respectively. The mass of X is 0.17 times the mass of Y. The value of g on Y is 9.0 m/s².

(a) What is the ratio of the mean density of X to that of Y?

ρ_X

ρ_Y

=

(b) What is the value of g on X?
m/s²

(c) The mass of Y is 1.349 ✕ 10²⁵ kg. What is the escape speed on X?
m/s

Answer 1

(a)

Radius of planet x,

Rx = 6.4*10^6 m / 2 = 3.2*10^6 m

Radius of planet y,

Ry = 2*10^7 m / 2 = 1*10^7 m

Volume of x / Volume of y = (4/3)*pi*(3.2*10^6 )^3 / (4/3)*pi*(1*10^7)^3

Vx / Vy = 0.0327

ratio of the mean density of X to that of Y,

x / y = (Mx / My) / (Vx / Vy)

x / y = (0.17) / (0.0327)

x / y = 5.18

(b)

Gravitational acceleration is given by,

g = GM / r^2

value of g on X = g on y * [(Mx / My) / (Rx / Ry)^2]

value of g on X = 9 * (0.0327) / (0.32)^2

value of g on X = 2.88 m/s^2

(c)

Escape speed is given by,

ve = sqrt (2GM / r)

escape speed on y = sqrt [(2*6.67*10^(-11)*1.349*10^25) / (1*10^7)]

escape speed on y = 13412.68 m/s

escape speed on X =  escape speed on y * sqrt [(Mx / My) / (Rx / Ry)]

escape speed on X = 13412.68 * sqrt [0.17 / 0.32]

escape speed on X = 9776.08 m/s