And to complete Chrono’s excellent explanation, the *energy *books remain balanced because whatever amount of energy the astronaut put into the bow which was wasted as heat when the bow was flexed & released was not imparted to the arrow and therefore didn’t affect the arrow’s velocity.
My astronauts are. It’s my example. I was simplifying intentionally.
But it is essentially incorrect in its particulars (e.g. an astronaut hovering stationarily) and is not related to the scenario of the o.p. or subsequent scenarios, which involve astronaut in orbit and the balance of forces that stems from that motion.
Stranger
Astronauts who are “hovering stationarily” are necessarily not in orbit. The two situations are mutually exclusive. Two astronauts “hovering stationarily” will not remain so-- they will immediately start falling toward the planet extremely slowly and will impact it eventually, absent other forces. Is that what you’re positing?
If they were. somehow, made stationary 100 miles above the planet, would they fall at the same rate they would from 20 feet up? How much has gravity decreased at such a distance from earth’s center of mass? Not counting air resistance, of course.
No, they would fall somewhat slower. More precisely, they would fall a shorter distance in the first second if they started at 100 miles up than they would at 20 feet. However, I don’t think that there would be much of a difference. Gravitational pull depends on distance from the center of mass, not the surface of an object. The Earth is almost 4,000 miles thick-- falling from 4,100 miles is going to be different but not much different. I’m having trouble digging up the equations to describe it, though.
Ah, another interesting question , as we both used the term “center of mass”. I see your point that the earth’s surface is 4,000 miles above the center, so 4,100 is not a real big difference.
Does gravity really work as if there is some theoretical point at the center from which it emanates? Is that just the way the forces sum up, or is there something else going on?
It does for spherical planets. Complex questions like the gravity on the inside of a doughnut require complex answers. Technically, the center of mass refers to the center of mass of the entire astronaut-Earth system. However, the astronaut is too small to bother with for normal calculations.
To clarify a bit on applecider’s good post.
A diffuse symmetrical mass acts like a point source just as a result of the way the forces sum up. Nothing else magic is going on.
For these kinds of high school physics problems we can pretend the Earth’s mass is symmetrical. In reality there are small “high” & “low” spots in the gravity field due to the lumpiness of the interior of the Earth. The fact the Earth is a slightly squashed almost-sphere also means that anything not orbiting directly along the Equator is moving through an elliptical gravity well, not a spherical one.
So the general case for orbits of more-or-or less spherical planets is a slightly lumpy eliptical gravity well.
And the further away from the Earth you get, the less and less the high and low spots and even the ellipticity matter when calculating the effect of Earth’s gravity on an object orbiting it (or on any object for that matter) - the high and low spots have important effects on satellites in low earth orbit, but not on the Moon and likewise, while the ellipticity of the Earth matters to the Moon, it doesn’t affect Mars much at all.