For this problem, we assumethat we are on Planet-I. The radius of this planet is

Question

For this problem, we assumethat we are on Planet-I. The radius of this planet isR =4740 km, thegravitational acceleration at the surface is gI =7.4 m/s2, and the gravitationalconstant G =6.67×1011N m2/kg2 in SI units. The mass of Planet-I is not given. Not all thequantities given here will be used. Suppose a cannon ball ofmass m = 5520 kg is projected vertically upward from thesurface of this planet. It rises to a maximumheight h =10001.4 km above the surface ofthe planet. Caution: Here thegravitational acceleration decreases as the cannon ball travelsaway from Planet-I.

Explanation / Answer

The radius of planet-1 is R = 4740 km = 4740 * 103m The gravitational acceleration at the surface of planet-1 isg1= 7.4 m/s2 The gravitational constant is G = 6.67 * 10-11Nm2/kg2 The mass of the cannon ball be m = 5520 kg The cannon ball rises to a maximum height h =10001.4 km = 10001.4 * 103 m  above thesurface of the planet. Let the mass of planet-1 be M Let the mass of planet-1 be M The acceleration due to gravity on the surface of planet-1is g1= (GM/r) or M = (g1* r/G) Here,r = R + h = 4740 * 103 + 10001.4 *103 = 14741.4 * 103 m Substituting the values in the above equation,we get M = (7.4 * 14741.4 * 103/6.67 *10-11) or M = 1.63 * 1018 kg The gravitational potential energy of the cannon ball as itrises to a maximum height h above thesurface of the planet is U = (GMm/r) ------------------(1) The gravitational potential energy of the cannon ball is equalto the kinetic energy of the cannon ball,that is, K = (1/2)mv2 ------------------(2) From equations (1) and (2),we get U = K or (GMm/r) = (1/2)mv2 or v = (2GM/r)1/2 or v = (2 * 6.67 * 10-11 * 1.63 *1018/14741.4 * 103)1/2 or v = 3.84 m/s Therefore,the speed of the cannon ball when it is projectedvertically upward from the surface of the planetis v = 3.84 m/s. or v = (2 * 6.67 * 10-11 * 1.63 *1018/14741.4 * 103)1/2 or v = 3.84 m/s Therefore,the speed of the cannon ball when it is projectedvertically upward from the surface of the planetis v = 3.84 m/s.

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