please exampble b and complete c The mean diameters of X and Y, two planets in t

Question

please exampble b and complete c

The mean diameters of X and Y, two planets in the same solar system, are 6.2 × 103 km and 1.2 × 104 km, respectively. The mass of X is 0.16 times the mass of Y. The value of g on Yls 8.6 m/s2 (a) what is the ratio of the mean density of ox = 1.16 PY to that of Y? (b) What is the value of g on X7 5.15 m/s2 (c) The mass of Y is 4.642 × 1024 kg, what is the escape speed on x? 179 m/s Density is the ratio of mass to volume. Do you recall how to calculate the volume of a sphere? The gravitatlonal acceleration is the ratlo of the gravitational force on a particle to the particle's mass. The escape speed is the speed when the escape kinetic energy equals the magnitude of the initial gravitational potential energy Additional Materials Section 13.5

Explanation / Answer

Part (a), you have already solved. So, I am solving part (b) and part (c).

Part (b) -

Acceleration due to gravity is -

g = M/R^2

therefore, the ratios are -  

g(X)/g(Y) = M(X)/M(Y)*(R(Y)/R(X))^2 = 0.16*[(1.2 x 10^4) / (6.2 x 10^3)]^2 = 0.60  

therefore, the value of g on X = 0.60*8.6 m/s^2 = 5.16 m/s^2

Part (c) -  

The expression of escape velocity is -

vesc = sqrt[2GM/R]

ratios:

vexc(X)/vesc(Y)= sqrt[(M(x)/M(Y) * R(Y)/R(X)] = sqrt[0.16*(1.2 x 10^4 / 6.2 x 10^3)]= 0.56

so -

vesc(X) = 0.56 * vesc(Y)

now -

vesc(Y) = sqrt[2GM(Y)/R(Y)]

= sqrt[(2x6.674x10^-11x4.642x10^24) / (1.2 x 10^7)] = sqrt[51.634 x 10^6] = 7.186 x 10^3 m/s

Therefore, escape velocity of X = 0.56 x 7.186 x 10^3 = 4.024 x 10^3 m/s

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