Zero, a hypothetical planet, has a mass of 2.0 times 10^23 kg, a radius of 3.0 t
ID: 1446465 • Letter: Z
Question
Zero, a hypothetical planet, has a mass of 2.0 times 10^23 kg, a radius of 3.0 times 10^6m, and no atmosphere. A 18 kg space probe is to be launched vertically from its surface. 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 from the center of Zero? 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
(a)
Since the gravitational force is a conservative force, the mechanical energy of the probe-planet
system is conserved. The energy conservation equation yields
Ei = Ef Ki – GMm/Ri = Kf – GMm/Rf Kf = Ki – GMm(1/Ri – 1/Rf)
where Ki = 5 * 10 7 J ; M = 2 * 1023 kg ; m = 18 kg ; Ri = 3 *106 m ; Rf = 4 *106 m ; G = 6.674 *10-11 N·m²/kg²
Kf = 2.99 * 107 J = 3 * 107 J
(b) When the probe reaches the maximum height, its speed becomes zero. The energy conservation
equation leads to
Ei = Ef Ki – GMm/Ri = -GMm/Rf
Ki = GMm(1/Ri – 1/Rf)
where M = 2 * 1023 kg ; m = 18 kg ; Ri = 3 *106 m ; Rf = 8 *106 m ; G = 6.674 *10-11 N·m²/kg²
Ki = 5.00 * 107 J
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