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Unformatted text preview: —1— Problem. Escape velocity As you know, Isaac Newton was trying to ﬁgure out how the moon goes around when an apple fell nearby. But do you know what he saw between the branches of the apple tree when he looked up? You’ll ﬁgure it out. Anyway his train of thought became something like “moon, moon, moon, apple, moon, apple gravity, moon, apple gravity, moon, gravity moon..., Oh!” Until then nobody knew gravity had anything to do with the motion of objects in space. They had thought that gravity was what made their physics book heavy. Eventually, Newton decided that the force between objects of masses M and m at a distance r apart is F = GM mr−2 where G = 6.67 × 10−11 m3 /( kg s2 ). a) As we already know, the integral r12 F dr represents the work associated with the force F . More explicitly, it gives the work done in moving m in M ’s gravity ﬁeld from r1 to r2 . Consider the following three cases: Case 1. r1 = 0 and r2 = r0 . Case 2. r1 = r0 > 0 and r2 = ∞. Case 3. r1 = 0 and r2 = ∞. For each case set up the work integral, determine if it is proper or improper, and for the improper ones, determine if they are divergent or convergent. If an integral is convergent, ﬁnd its value.
r —2— b) Using case 2, ﬁnd the “escape velocity” from Earth, i.e., the minimal launching velocity needed for a rocket to leave the gravitational ﬁeld of the Earth. Hint: Here we take “escaping from Earth” to mean that the rocket’s distance from it will tend to inﬁnity as t → ∞; hence r2 = ∞, as in case 2 above. Assume that the energy is to be supplied (by rocket engines) in the form of initial kinetic energy, 1 mv 2 , where v is the velocity. The Earth’s radius is r0 = 6378 km and it has an 2 average mass density of ρ = 5500 kg/m3 . If you want to learn more about problems of this kind, consider taking the following engineering course: TAM 2030 Dynamics (also ENGRD 2030). ...
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