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?
To determine the initial kinetic energy required for the space probe to achieve a maximum distance of 8.0x10^6 m from the center of Zero, you can use the conservation of mechanical energy. The sum of the initial kinetic energy (KE) and potential energy (U) should equal the sum of the final kinetic energy and potential energy at the maximum distance.
The equation you can use is:
(1/2) mV^2 - GMm/R + mgh = KE + U
In this case, we'll take the reference point for potential energy (U) as the surface of Zero, so h is the height above the surface (8.0x10^6 m). Here's how you can calculate it:
1. Calculate the escape velocity (Ve) using the formula:
Ve = sqrt(2GM/R)
2. Use the escape velocity (Ve) to calculate the initial potential energy (U0) as the probe starts from the surface of Zero:
U0 = -GMm/R
3. Calculate the potential energy (U) at the maximum distance (8.0x10^6 m):
U = -GMm/(R + h)
4. Set up the equation using the conservation of mechanical energy:
(1/2) mV^2 + U0 = (1/2) m(Ve)^2 + U
5. Rearrange the equation to solve for the initial kinetic energy (KE0):
KE0 = (1/2) m[(Ve)^2 - V^2] + (U - U0)
Plug in the values:
Ve = sqrt(2GM/R) = sqrt(2(6.67e-11)(1.0e23)/(3.0e6)) ≈ 2108.7 m/s
V = initial velocity at the surface of Zero, which is also equal to the escape velocity (Ve)
U0 = -GMm/R = -6.67e-11 * 1.0e23 * 10 / 3.0e6 ≈ -2.22e8 J
U = -GMm/(R + h) = -6.67e-11 * 1.0e23 * 10 / (3.0e6 + 8.0e6) ≈ -1.11e8 J
KE0 = (1/2) * 10 * [(2108.7)^2 - (2108.7)^2] + (-1.11e8 - (-2.22e8)) = -1.11e8 - (-2.22e8) = 1.11e8 J
Therefore, the initial kinetic energy required for the space probe to achieve a maximum distance of 8.0x10^6 m from the center of Zero is approximately 1.11x10^8 J.