The Formula you use is the correct one.
However, consider this. If the initial PE existed, wouldn't you have to add it to the initial KE then subtract the PE at some R from the center? We normally make ground zero PE level, but for this...
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!
How about the equation:
KE(initial)-(GMm)/R = KE(final)-(GMm)/10R
I am still confused as to what value to use for R for a...Do I use 4e6 for both R's?
Thanks.
To solve this problem, we can start with the conservation of mechanical energy, which states that the initial mechanical energy (sum of kinetic energy and potential energy) is equal to the final mechanical energy.
For part (a), we can start with the initial kinetic energy given as 5.0x10^7 J. As there is no initial potential energy (since ground level is taken as zero), the initial mechanical energy is just the initial kinetic energy.
To find the final kinetic energy when the probe is 4.0x10^6 m from the center, we need to calculate the final potential energy at that distance. The potential energy equation for an object at a certain distance R from the center of a planet is PE = -GMm/R, where G is the gravitational constant, M is the mass of the planet, m is the mass of the object, and R is the distance from the center of the planet.
The equation you mentioned, E = KE - GMm/R, is not the correct form. Instead, we can set up the equation for conservation of energy as follows:
Initial kinetic energy + initial potential energy = final kinetic energy + final potential energy.
So, we have:
(5.0x10^7 J) + 0 = final kinetic energy + final potential energy
Since the probe is 4.0x10^6 m from the center at the final position, we can calculate the final potential energy as -GMm/R. The mass of Zero is given as 1.0x10^23 kg, and the mass of the probe is 10 kg. Thus, we have:
Final potential energy = -(6.67x10^-11 Nm^2/kg^2) * (1.0x10^23 kg) * (10 kg) / (4.0x10^6 m)
Now, we can rearrange the equation to solve for the final kinetic energy:
Final kinetic energy = (5.0x10^7 J) - final potential energy
Substituting the value of the final potential energy and carrying out the calculation will give you the final kinetic energy.
For part (b), we can start with the same equation for conservation of energy, but this time the final kinetic energy will be zero at the maximum distance from the center.
So, we have:
Initial kinetic energy + initial potential energy = 0 + final potential energy
Given that the maximum distance is 8.0x10^6 m, we can calculate the final potential energy by substituting the values into the potential energy equation.
Remeber that for both parts (a) and (b) you need to use the correct values for R, which are 4.0x10^6 m for part (a) and 8.0x10^6 m for part (b). This is because R represents the distance from the center of Zero at the respective positions.
Hope this helps clarify the approach to solving the problem. Let me know if you need further assistance!