Zero, a hypothetical planet, has a mass of 1.0 1023 kg, a radius of 3.0 106 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 5.0 107 J, what will be its kinetic energy when it is 4.0 106 m from the center of Zero?
(b) If the probe is to achieve a maximum distance of 8.0 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?
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What equation do I start with?
I tried using:
E = KE - GMm/R and came up with the wrong answer. Am I doing something wrong? PLEASE HELP!
To solve this problem, you can use the law of conservation of energy. The equation you mentioned, E = KE - GMm/R, is a valid starting point. However, there might be an issue with the values of the variables used in your calculation. Let's break down how to approach each part of the problem:
(a) To find the kinetic energy of the space probe when it is 4.0 x 10^6 m from the center of Zero, you need to use the conservation of energy. The formula is:
E = KE + PE
where E is the total mechanical energy, KE is the kinetic energy, and PE is the potential energy.
At the surface of Zero (r = 3.0 x 10^6 m), the potential energy is zero because we are using the surface as the reference point. So the total mechanical energy at the surface is just the kinetic energy:
E = KE
Using this equation, the initial total mechanical energy is 5.0 x 10^7 J.
As the probe moves away from the center of Zero to a new position 4.0 x 10^6 m away, it will gain potential energy and lose an equal amount of kinetic energy, assuming no external forces act upon it. Therefore, we can write:
E = KE + PE
E = KE1 + PE1
where KE1 is the initial kinetic energy and PE1 is the initial potential energy.
At the new position, the potential energy is given by the equation:
PE1 = - GMm/r1
where G is the gravitational constant, M is the mass of Zero, m is the mass of the probe, and r1 is the new distance from the center of Zero.
Now, to find the kinetic energy at the new position, we can rearrange the equation:
E = KE1 + PE1
KE1 = E - PE1
Substituting the given values and solving for KE1 will give you the answer.
(b) To find the initial kinetic energy required for the probe to achieve a maximum distance of 8.0 x 10^6 m from the center of Zero, you need to use the same principles as in part (a). The difference is that, in this case, you need to find the potential energy at the maximum distance instead of a given distance.
The potential energy at a distance r2 for the probe is given by:
PE2 = - GMm/r2
Using the conservation of energy equation:
E = KE2 + PE2
KE2 = E - PE2
Substitute the given values and solve for KE2.
Remember to use the correct values for the gravitational constant, mass of Zero, and mass of the probe in your calculations. With the correct values and equations, you should be able to find the correct answers for both parts of the problem.