This description of orbits seems wrong to me

[Doug1943]

and I would be grateful if those who understand classical mechanics, in particular the physics of orbital motion, could visit this link -- https://earthkam.ucsd.edu/resources/students_orbital_mechanics#shuttle_stays_in_orbit -- and read the description of the 'four cases' by which a spacecraft may stay in orbit. I must be misunderstanding something, but it seems nonsensical to me -- cases A and B show it crashing to the ground and not achieving orbit at all (and cases C and D also seem wrong, for different reasons).

 

[Drakkith]

Yeah, it's really badly worded on B, while A is just wrong. C and D look fine to me, though.

 

[Bandersnatch]

The wording is bad, especially in A which makes sense only if one thinks of Earth's centre of mass rather than the globe of the planet. But then any speed is "enough speed".

Giving it the benefit of the doubt, I think the approach is to explain that all cases are technically orbits(around the centre of mass of the Earth), only in A and B the orbits intersect with the Earth's surface.
In that sense, the ballistic path of any projectile(e.g., rocks, bullets) is a section of an orbit.

IMO, the proper way would be to merge A and B together(there's no qualitative difference), explain that all flight paths are ellipses with Earth's centre of mass in one of the foci, and that the velocity determines whether the periapsis will be above or below the surface. Circular orbit to be mentioned as a special case.

 

[Doug1943]

Well, yes ... I suppose all cases of bodies in motion are, if we don't abstract away anything at all, actually cases of many-body paths, and not even ellipses or any other conic section, which assume just two centers of mass. But we usually abstract out enough to allow us to focus on interesting special cases: thus rifle bullets and baseballs follow, in our consideration of their paths, parabolas, although 'really' their paths are not quite parabolas, even after removing the effect of air resistance. And it seems perverse to use the word 'orbit' for a path that does not actually ... orbit.

And note that this site is supposed to be a site to get school children interested in science. (Looking at a previous section, I suspect that the task of writing up this section of the site got given to an over-zealous graduate student.)

 

[D H]

That website provides (or rather, attempts to provide) a description of orbits along the lines of Newton's cannon. It is admittedly a poor description.

Imagine an airless, non-rotating planet with a single tall mountain. Imagine a cannon emplaced atop that mountain. Imagine that one can increase the amount of charge (gunpowder) placed in the cannon, without limit.

With a tiny amount of charge, the cannonball will follow what appears to be a parabolic arc and eventually land somewhere on the mountain. If you keep increasing the charge with each shot, eventually the cannonball will fall clear of the mountain and hit somewhere on the featureless plain out of which the mountain arises.

Increase the charge even more and you can no longer pretend that featureless plain is a plane. It's the surface of a curved planet. The path followed by the cannonball no longer looks like a parabolic arc. It instead looks like an elliptical arc. If you look back, that supposed parabolic arc that the cannonball followed with a lesser charge is actually an elliptical arc as well. Increase the charge even more an the cannonball strikes the planet a quarter way round the planet from the mountain, then 3/8 the way around, and eventually, just shy of half way around.

Now when you increase the charge something rather different happens. The cannonball's altitude decreases at first, but when it reaches the antipode (halfway around point), the altitude starts increasing. The cannonball does not hit the planet. Instead it falls all of the way around the planet and will eventually hit the cannon if you don't move it out of the way. The cannonball is in orbit!

If you keep increasing the charge you will find that the altitude at the antipode keeps increasing. Increase the charge enough and the cannonball will maintain the same altitude throughout its flight. Increase it even more and the height at the antipode exceeds the height of the mountain + cannon.

At this point you'll notice something else happening. With every increase in charge, it takes the cannonball longer and longer to make its round the planet trip. Increase the charge even more and the cannonball won't return, ever. The path now is parabolic, with the cannonball heading off to infinity rather than following a closed path. Increase the charge even more and the path becomes hyperbolic.