Zero, a hypothetical planet, has a mass of $5\mathrm{.}5{10}^{23}kg,$ a radius of $2\mathrm{.}7{10}^{6}m,$ and no atmosphere. A $10kg$ space probe is to be launched vertically from its surface. $($a$)$ If the probe is l$$unched with an initial kinetic energy of $5\mathrm{.}0{10}^{7}J,$ what will be its kinetic energy when it is $4\mathrm{.}0{10}^{6}m$ from the center of Zero? $($b$)$ If the probe is to achieve a maximum distance of $8\mathrm{.}0{10}^{6}m$ from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

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