Posted by Technoboi11 on Wednesday, March 7, 2007 at 1:09am.
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?
(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?
For Further Reading
* Physics/Math - drwls, Wednesday, March 7, 2007 at 5:55am
Use the rel;ationship
KE + Potential energy) = constant.
The potential energy at distance R from the center is
-GMm/R
M is the planet's mass and m is the probes.
That means (1/2) mV^2 - GMm/R = constant
Use that fact to compute the unknown kinetic energy in each problem
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this is what i did..
V = sqrt(2GM/R)
V = sqrt(2(6.67e-11)(1.0e23)/(3.0e6))
V = 2108.7
[..would i use earth's gravity for this question? .. if not.. what should i use for G?]
then..
(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
.. what am i doing wrong?
and how would i go about with the 2nd question?
please help!! thanks! :)
To answer the first question, you are using the correct formula for potential energy and the relationship between kinetic energy and potential energy. However, you made a mistake in the calculation of the potential energy term.
The correct formula for potential energy at distance R from the center is -GMm/R, where G is the gravitational constant (6.67x10^-11 N m^2/kg^2), M is the mass of the planet (1.0x10^23 kg), and m is the mass of the probe (10 kg). Therefore, the potential energy term should be -GMm/R = -(6.67x10^-11)(1.0x10^23)(10)/(4.0x10^6).
Now, let's correct your calculation:
KE + Potential energy = constant
(1/2)mV^2 - GMm/R = constant
(1/2)(10)(V^2) - (6.67x10^-11)(1.0x10^23)(10)/(4.0x10^6) = constant
Since you know the initial kinetic energy is 5.0x10^7 J, you can substitute that into the equation and solve for V:
(1/2)(10)(V^2) - (6.67x10^-11)(1.0x10^23)(10)/(4.0x10^6) = 5.0x10^7
Now, simplify the equation and solve for V:
5V^2 - (6.67x10^-11)(1.0x10^23)(10)/(4.0x10^6) = 5.0x10^7
5V^2 = 5.0x10^7 + (6.67x10^-11)(1.0x10^23)(10)/(4.0x10^6)
Once you solve for V, you can calculate the kinetic energy of the probe when it is 4.0x10^6 m from the center of Zero using the formula KE = (1/2)mv^2.
Now let's move on to the second question. 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 use the same equation: KE + Potential energy = constant. However, this time the potential energy term will be -GMm/R, where R is 8.0x10^6 m.
You can set up the equation as follows:
(1/2)mV^2 - GMm/R = constant
(1/2)(10)(V^2) - (6.67x10^-11)(1.0x10^23)(10)/(8.0x10^6) = constant
Since you want to find the initial kinetic energy (KE), you can set the constant to 0. The equation then becomes:
(1/2)(10)(V^2) - (6.67x10^-11)(1.0x10^23)(10)/(8.0x10^6) = 0
Now, solve for V and calculate the corresponding kinetic energy using the formula KE = (1/2)mv^2.
I hope this explanation helps you understand how to approach these problems. Let me know if you have any further questions!