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?
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* 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 solve the first part of the problem, you did most of the calculations correctly. However, you made a small mistake in the expression inside the parentheses in the potential energy term. It should be minus (GMm)/R^2, not minus (GMm)/R. The correct expression would be:
(1/2) mV^2 - GMm/R^2 = constant
Now, let's solve it again using the correct expression:
V = sqrt(2GM/R)
V = sqrt(2(6.67e-11)(1.0e23)/(3.0e6))
V = 2108.7 m/s
Now, substitute the values into the energy equation:
(1/2) mV^2 - GMm/R^2 = constant
(0.5)(10)(2108.7)^2 - (6.67e-11)(1.0e23)(10)/(4.0e6)^2
= 98142857 - 20875000
= 77267857 J
Therefore, the kinetic energy of the probe when it is 4.0x10^6 m from the center of Zero is 7.7x10^7 J.
For the second part of the problem, we can use the same energy equation. We need to find the value of the initial kinetic energy, given the maximum distance of 8.0x10^6 m from the center of Zero.
Using the same formula for V:
V = sqrt(2GM/R)
V = sqrt(2(6.67e-11)(1.0e23)/(8.0e6))
V = 1781.9 m/s
Now, substitute these values into the energy equation:
(1/2) mV^2 - GMm/R^2 = constant
(0.5)(10)(1781.9)^2 - (6.67e-11)(1.0e23)(10)/(8.0e6)^2
= 63685286 - 15687500
= 47997786 J
Therefore, to achieve a maximum distance of 8.0x10^6 m from the center of Zero, the probe must be launched from the surface with an initial kinetic energy of approximately 4.8x10^7 J.