Zero, a hypothetical planet, has a mass of 5.9 times 10^23 kg, a radius of 3.4 t

Question

Zero, a hypothetical planet, has a mass of 5.9 times 10^23 kg, a radius of 3.4 times 10^6 m, and no atmosphere. A 10 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? (a) Number: 2.91E7 Units: J (b) Number: 5.0E7 Units: J

Explanation / Answer

Gravitational PE = -G m1 m2 / r

For initial PE: r1 = R

PEi = - G M m / R

KEi = 5 x 10^7 J

PEf = - G M m / (R + r)

Applying energy conservation,

PEi + Kei = PEf + KEf

- (6.67 x 10^-11 x 5.9 x 10^23 x 10 / 3.4 x 10^6) + (5 x 10^7) = -(6.67 x 10^-11 x 5.9 x 10^23 x 10 / 4 x 10^6) + KE

- 11.6 x 10^7 + 5 x 10^7 = - 9.8 x 10^7 + KE

KE = 3.2 x 10^7 J .............Ams

(B) at maximum distance , KE = 0

so KEf = 0

Applying energy conservation,

- 11.6 x 10^7 + KEi = KEf - (6.67 x 10^-11 x 5.9 x 10^23 x 10 / 8 x 10^6)

- 11.6 x 10^7 + KEi = 0 - 4.9 x 10^7

KEi = 6.7 x 10^7 J ........Ans

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