Now imagine an average man of 70kg floating in space with no gravitational forces acting on him. Now a pebble(25 grams) spawns into existence near him. So now there's a gravitational force between them. Now, I know that gravity won't be uniform across the man's body but let's assume that it is uniform around his waist and the pebble is just orbiting the man's waist. I would like to know what would be the velocity needed for the pebble to escape the gravitational pull and move away.
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1$\begingroup$ What is the gravitational attractive force as a function of distance? Now integrate to infinity to get the energy needed to get there. Now what is the escape velocity? $\endgroup$– Jon CusterCommented Feb 25, 2023 at 15:13
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$\begingroup$ @JonCuster How exactly do i do it? Sorry I'm no physics expert, just an amateur student. $\endgroup$– Electric-BasketCommented Feb 25, 2023 at 16:37
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1$\begingroup$ First, assume a spherical cow, er, human … $\endgroup$– DarkDustCommented Mar 1, 2023 at 10:29
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1 Answer
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Escape velocity from a spherical body is given by
$$v_{esc} = \sqrt {\frac {2GM}{r}} $$
Where $G$ is the gravitational constant ≈ $6.67×10^{-11} m^{3}·kg^{−1}·s^{-2}$, $M$ the mass of the body orbited (technically, the sum of the masses of the orbiting and orbited bodies, but usually the secondary mass is negligible), and $r$ the distance from the center of mass. For a 70 kg body orbited at a distance of, say, 50 cm, Wolfram Alpha tells me this works out to about 136 µm per second, or 492 mm/hour.
Note that escape velocity is $\sqrt 2$ times the circular orbit velocity, so the pebble could orbit the man at a lazy 348 mm/hour, completing an orbit every nine hours or so.
The gravitational force between them would be about 1 micro-Newton.
answered Feb 25, 2023 at 16:21
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$\begingroup$ I see. Thanks for the answer, it was really helpful. $\endgroup$ Commented Feb 25, 2023 at 16:25
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$\begingroup$ do you know if the "lumpiness" of a human body actually changes escape velocity? I know it affects the stability of orbits. $\endgroup$ Commented Feb 25, 2023 at 21:21
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$\begingroup$ Yeah, for such a close orbit there would definitely be an effect. I think if you modeled the body as a cylinder and orbited radially around it, the effective mass would be reduced (because some of it would be farther from the orbiter), and thus the orbital and escape velocities would be lower. For an orbiter say 100 meters from the body the shape would have little impact to escape velocity. $\endgroup$ Commented Feb 25, 2023 at 21:45
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$\begingroup$ @Electric-Basket If this answer helped you, please consider accepting it by clicking the checmark. $\endgroup$ Commented Feb 26, 2023 at 6:33