So if you want to be really pedantic, it's never an ellipse because the Earth is not a point mass. It would be equivalent to a point mass if the Earth were a perfect sphere of uniform density, but it isn't. In reality it's a potato like mass blob that's approximated by what geodesists call the "geoid". So in order of approximations the path of a ball thrown on earth is a parabola -> ellipse -> numerical integration of 6-dof initial conditions and spherical harmonics approximation of the earth gravity field.
This turns out to be not pedantic but very important if you're guiding an ICBM. And when landing on the Moon, Apollo had to deal with irregularities in the Moon's gravity due to mass concentrations, called mascons.
(If you're interested in missile guidance, take a look at the book Inventing Accuracy. Among other things, it discusses some of the efforts to map the Earth's gravity field to increase missile accuracy for Trident and Minuteman missiles. I knew a physicist who worked on this.)
It wouldn't be an ellipse even if Earth were a point mass. The gravity of the Moon and Sun, the gravitational lumpiness in the sky, has the same effect as the gravitational lumpiness underground. The combined result may be no closer to an ellipse than it is to a parabola.
Source is a table in the first chapters of Fundamentals of Astrodynamics.
It's not a force. From the right frame of mind[0], it's just a straight line.
[0] Substituting for frame of reference.
ICBMs were never used to toss nukes at people.
Pretty sure their existence saved the world from WW3.
> Surely there's something else we could be doing instead?
The application most useful in everyway life is GPS.
So, moral of the story - we have to be not too pedantic.
I'm not a physicist, but that doesn't sound right. One example is that if you are inside of the sphere, at least some of the mass will be pulling you away from the point at its center. I think what you mean is that a point mass closely approximates a spherical mass, but the degree to which that is true becomes less and less the closer you get to the center. I don't think you have successfully contradicted what the person you responded to said.
Proving this is a classic problem in undergraduate physics.
If the mass is spherically symmetric, this will not be the case; all of the Newtonian forces from the masses further away from the center than you are will cancel out. This is called the "shell theorem", and it turns out to hold even in General Relativity.
For each point of any given distance, you can find a disk of points at the plane of the same distance whose gravitational pull will be equivalent to a single point at their center.
You do this for all distances x, and you will find an equivalent rod going from the attracted objected to the center of the sphere, of a non-uniform but symmetrical density.
We find that the density is linearly correlated to the area of the corresponding section, and we then take the closer half of the rod, multiply the density by 1/r^2, and find that it is constant!
For the far half of the rod, we cannot do this, so instead we divide it into two halves of equal pull, then divide those to halves and so on, and find that it this reduces to the equivalent of a point.
Now that we have two points in-line with the object, we find the point with the same total mass that exerts and equivalent force, and lo and behold it is at the center.
I'm sure there is a much simpler way to intuit it, though :)
Would it? Even of uniform density, the mass would not the at the core, only the center of mass would be. Most of the mass is actually closer to the surface. I'm no physicist, but I'd imagine that would have quite some impact[0] on any trajectory that crosses the surface.
And for any ball thrown at human speeds, I'd expect Earth's gravity would be much closer to a uniform gravitational field than one from a point mass.
[0] Pun not originally intended but I'm quite happy with it now.
In relativity, there is no such thing as a "uniform gravity field", if by that you mean a field where the "acceleration due to gravity" is the same everywhere. The closest you can come is the "gravity field" inside a rocket accelerating in a straight line in empty space, where the acceleration felt by the crew is constant. That kind of "gravity field" has an "acceleration due to gravity" that decreases linearly with height.
Look up the Bell Spaceship Paradox. In relativity, two spatially separated objects (or two ends of a spatially extended single object) that have the same proper acceleration in the same direction do not stay at rest relative to each other; they move apart, as seen in each of their own frames. This is different from the behavior predicted by Newtonian mechanics. In order to have the two objects (or two ends of a single extended object) remain at rest relative to each other, the one in front has to have a smaller proper acceleration than the one in back. (Rindler coordinates are often used to describe this case.)
Even if you consider the point mass to move because of the mutual attraction?
Based on what I've seen over the years I've been around here, I'm quite certain that's a primary motivator for a non-trivial number of commenters here.
At least classically, a sphere is indistinguishable (gravitationally) from a point mass while you're outside it. The earth is pretty sphere-ish, locally speaking.
Kinda. There are mountains, they have their own gravitational attraction, and it can be measured... even in the 18th century! https://en.m.wikipedia.org/wiki/Schiehallion_experiment
It all depends on how pedantic one wants to be :)
