Jupiter has numerous moons. Four moons were discovered by Galileo in 1610. One o

Question

Jupiter has numerous moons. Four moons were discovered by Galileo in 1610. One of these moons, Ganymede, is bigger than Mercury. Ganymede has a mass of 14.8x1022 kg, a period of 7.16 earth days, and a mean distance from Jupiter of 1.07x106 km. Your TA's have recently discovered a new moon to Jupiter, and appropriately named it the Lost Ball Orbiter (LBO). This moon has a mean distance from Jupiter of 12.8 x106 km, a radius of 2600 km, and a density of 2000 kg/m3. LBO follows an elliptical path with e=0.04.

a.) What is the period of LBO?

b.) What is the closest LBO comes to Jupiter? (106 km)

c.) What is the gravitational acceleration on LBO?

d.) What is the escape velocity for a spaceship leaving LBO?

Explanation / Answer

a)

The time period of the of LBO is

T =2pi*r3/2/Sqrt(GMjupiter) =2*3.14*(12.8*109)3/2/Sqrt[(6.67*10-11)(1.8986*1027)

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b)

The closest distance is given by

rp =r(1-e) =(12.8*106km)(1-0.04)=12.288*106km

c)

The gravitational acceleration of the LBO is

g =GM/R2 =Gp(4/3*pi*R3)/R2 =4*G*pi*p*R/3

=4(6.67*10-11)(3.14)(2000)(2.6*106)/3=1.452m/s2

d)

The escape velocity is

ve =Sqrt(2gR) =Sqrt(2*1.452*2.6*106)=2747.799m/s

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