Ok...I figured out part a but I am having trouble with b. part a was:
Zero, a hypothetical planet, has a mass of 1.0x10^23 kg, a radius of 3.0x10^6 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 5.0x10^7 J, what will be its kinetic energy when it is 4.0x10^6 m from the center of Zero?
so I did KE + U = constant,
V = sqrt(2GM/R)
V = sqrt(2(6.67e-11)(1.0e23)/(3.0e6))
V = 2108.7
(1/2) mV^2 - GMm/R = constant
(.5(10)(2108.7)^2 - ((6.67e-11)(1.0e23)(10)/(4.0e6))
111165392.3 - 16675000 = 94490392.3 = 9.4e7=constant
9.4e7-5e7=4.4e7 J which is the right answer...but for b:
If the probe is to achieve a maximum distance of 8.0x10^6 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?
->I am not sure what to do. I recalculated the constant using the radius of 8e6 m and got 2.78e6 J but then I am not sure what to use for U. U=mgd so I tried U=(10)(9.8)(3e6) and subtracted that from 2.78e6 J but that is wrong. What am I missing?

Potential energy is not M g d in an inverse-square-law gravitational field. It is -MmG/r = -MmG/R^2 * R^2/r
= -m g'R^2/r
where g' is the acceleration of gravity at the planet's surface, M is the planet's mass, and R is the planet's radius. m is the mass of the probe and r is the distance of the probe from the center of the planet. That is probably where you made your mistake

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  1. my book has Mass=5.0 X 10^(23)

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