The mean diameters of planets A and Bare 7.7 × 10 3 km and 1.5 ×10 4 km, respect

Question

The mean diameters of planets A and Bare 7.7 × 103 km and 1.5 ×104 km, respectively. The ratio of the mass ofplanet A to that of planet B is0.19. (a) What is the ratio of the meandensity of A to that of B? (b) What isthe ratio of the gravitational acceleration on A to that onB? (c) What is the ratio of escape speedon A to that on B?

I can get parts (a) and (b) justfine but can't get (c) for some reason. Any help would begreatly appreciated!
I can get parts (a) and (b) justfine but can't get (c) for some reason. Any help would begreatly appreciated!

Explanation / Answer

Given , The mean diametersof planets A and B are   D = 7.7 ×103 km
                                                               and d = 1.5 × 104 km, So, the mean radius of the planets   A abdB are R = D / 2 = 3.85 * 10 ^ 3 km                                                                             r = d / 2 = 0.75 * 10 ^ 4 km Mass of A / mass of B = 0.19 So, mass of A M = 0.19 m where m = mass ofplanet B ( a) . ratio of mean density of A to that of B = ( M / V) / ( m / v ) Since density = mass / volume where V = volume of A = ( 4/3 ) R ^ 3            v= volume of B = ( 4/ 3 ) r ^ 3 substitue values weget answers (b). the ratio of the gravitational acceleration on A to thaton B = g / g ' where g =  gravitational acceleration on A =GM / R ^ 2            g' =  gravitational acceleration on B = Gm / r ^ 2 from this g / g ' = ( r / R ) ^ 2 * ( M / m) (3). escape speed v = [ 2Gm / R ] So, ratio of escape speed on A to that on B  = v / v ' here v = escape speed on A = [ 2GM / R ]        v ' = escape speed on B =[ 2Gm / r ] substitue values wegetanswer                                                              So, the mean radius of the planets   A abdB are R = D / 2 = 3.85 * 10 ^ 3 km                                                                             r = d / 2 = 0.75 * 10 ^ 4 km Mass of A / mass of B = 0.19 So, mass of A M = 0.19 m where m = mass ofplanet B ( a) . ratio of mean density of A to that of B = ( M / V) / ( m / v ) Since density = mass / volume where V = volume of A = ( 4/3 ) R ^ 3            v= volume of B = ( 4/ 3 ) r ^ 3 substitue values weget answers (b). the ratio of the gravitational acceleration on A to thaton B = g / g ' where g =  gravitational acceleration on A =GM / R ^ 2            g' =  gravitational acceleration on B = Gm / r ^ 2 from this g / g ' = ( r / R ) ^ 2 * ( M / m) (3). escape speed v = [ 2Gm / R ] So, ratio of escape speed on A to that on B  = v / v ' here v = escape speed on A = [ 2GM / R ]        v ' = escape speed on B =[ 2Gm / r ] substitue values wegetanswer                                                             

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