Zero, a hypothetical planet, has a mass of 2.0 times 10^23 kg, a radius of 3.0 t

Question

Zero, a hypothetical planet, has a mass of 2.0 times 10^23 kg, a radius of 3.0 times 10^6 m, and no atmosphere. A 17 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial kinetic energy of 5.0 times 10^7 J, what will be its kinetic energy when it is 4.0 times 10^6 m from the center of zero? (b) If the probe is to achieve a maximum distance of 8.0 times 10^6 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

Explanation / Answer

Using energy conservation:

Ei = Ef

Ki + PEi = Kf + PEf

PEi = GMm/Ri

PEf = GMm/Rf

Kf = Ki - GMm(1/Ri - 1/Rf)

Using the given values:

Kf = 5*10^7 - 6.67*10^-11*2*10^23*17*(1/(3*10^6) - 1/(4*10^6))

Kf = 3.11*10^7 J

B.

At the max height Kf = 0 J

So,

Ki = GMm*(1/Ri - 1/Rf)

Ki = 6.67*10^-11*2*10^23*17*(1/(3*10^6) - 1/(8*10^6))

Ki = 4.724*10^7 J

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