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?
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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:
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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
my book has Mass=5.0 X 10^(23)
To solve part b, you need to use the conservation of mechanical energy. The total mechanical energy of the system, which includes both kinetic energy (KE) and potential energy (U), remains constant throughout the motion.
The equation you need to work with is: KE + U = constant
To find the initial kinetic energy required for the probe to achieve a maximum distance of 8.0x10^6 m from the center of Zero, you need to determine the potential energy at this maximum distance and set it equal to the initial mechanical energy.
First, let's calculate the potential energy at a distance of 8.0x10^6 m from the center of Zero. The potential energy equation in an inverse-square-law gravitational field is:
U = -m * g' * (R^2 / r)
where m is the mass of the probe, g' is the acceleration of gravity at the planet's surface, which can be calculated using the equation g' = G * M / R^2, where G is the gravitational constant, M is the planet's mass, and R is the planet's radius. r is the distance of the probe from the center of the planet.
Substituting the values given:
g' = (6.67e-11 N*m^2/kg^2) * (1.0e23 kg) / (3.0e6 m)^2
g' = 3.7 m/s^2
U = -10 kg * 3.7 m/s^2 * (3.0e6 m)^2 / (8.0e6 m)
U = -1.384e9 J
Now, set the initial mechanical energy equal to the sum of the initial kinetic energy and potential energy at the maximum distance:
KE + U_initial = KE_initial + U_maximum
Since the probe is at rest initially, the initial kinetic energy (KE_initial) is zero:
U_initial = KE + U_maximum
Solving for KE, we get:
KE = U_initial - U_maximum
KE = 2.78e6 J - (-1.384e9 J)
KE = 1.386e9 J
Therefore, the probe must be launched with an initial kinetic energy of 1.386e9 J from the surface of Zero in order to achieve a maximum distance of 8.0x10^6 m from the center of Zero.