6 ms·

Off topic, but in reality, the trajectory of a ball thrown on Earth is not a parabola, but an ellipse [1]: > under the laws of gravity, a parabola is an imposs

Off topic, but in reality, the trajectory of a ball thrown on Earth is not a parabola, but an ellipse [1]: > under the laws of gravity, a parabola is an impossible shape for an object that's gravitationally bound to the Earth. The math simply doesn't work out. If we could design a precise enough experiment, we'd measure that projectiles on Earth make tiny deviations from the predicted parabolic path we all derived in class: microscopic on the scale of a human, but still significant. Instead, objects thrown on Earth trace out an elliptical orbit similar to the Moon. [1] https://www.forbes.com/sites/startswithabang/2020/03/12/we-all-learned-physics-biggest-myth-that-projectiles-make-a-parabola/ https://www.forbes.com/sites/startswithabang/2020/03/12/we-a...

Smaug123 6y ago
Good point; this becomes more obvious if you imagine throwing the ball up and then immediately collapsing all the mass of the Earth into a single point at the centre. What path does the ball follow now? It's probably following a path we would more usually call an "orbit", and it sure looks a lot like an ellipse. Now just put the mass of the Earth back where it was, and notice that the ball hits the ground before it can trace out very much of its orbit.
- est31 6y ago
  Despite both being conic sections, cutting up an ellipse won't yield you parabolas. An ellipse has two focal points to which the sum of the distances is constant, while a parabola has a focal point and a directrix line to which the difference of the distances is constantly 0. Two different things.
  - Koshkin 6y ago
    Isn’t a conic section made by cutting up, well, a cone?
  - srtjstjsj 6y ago
    More symmetrically, Every comic section has one focus and one directrix (and a semi-major axis for scale), eccentricity is the ratio of distance. Ellipse has eccentricity less than 1, and a parabola has eccentricity equal to 1.
  - pdkl95 6y ago
    Parabolas are just a special case of ellipses where one focus is infinitely far away. Vi Hart has a nice explanation: https://www.youtube.com/watch?v=v-pyuaThp-c https://www.youtube.com/watch?v=v-pyuaThp-c
dmurray 6y ago
A good point to make, but if one is going to be that pedantic one shouldn't call it "impossible". A parabola is possible with the right wind, air resistance, gravity from nearby mountains, etc, and in practice those have a larger impact on almost any suborbital projectile than the difference that turns the trajectory from a parabola to an elliptical arc.
saagarjha 6y ago
Speak for yourself. My balls reach escape velocity and then make corrections to assume e=1.
fouronnes3 6y ago
It's a parabola in a uniform gravity field, an ellipse in a circular gravity field coming from a point mass. 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.
- centimeter 6y ago
  > it's never an ellipse because the Earth is not a point mass. 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.
  - libraryofbabel 6y ago
    > 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 https://en.m.wikipedia.org/wiki/Schiehallion_experiment It all depends on how pedantic one wants to be :)
    - perl4ever 6y ago
      The first time I heard of the concept, it wasn't actually that, it was how in the 60s, they had to correct for mascons in the moon while orbiting.
      - garmaine 6y ago
        There are very few stable orbits close to the lunar surface. Basically a couple of polar orbits with very specific parameters, and that's it. The rest get so perturbed by gravitational anomalies that they fall out of orbit after a few months or years--faster than low Earth orbit where there is atmospheric drag! https://www.lpi.usra.edu/lunar/documents/NTRS/collection2/NASA_TP_3394.pdf https://www.lpi.usra.edu/lunar/documents/NTRS/collection2/NA...
gpsx 6y ago
If you want to get technical, what about air resistance?
ummwhat 6y ago
Ellipse, parabola, hyperbola and circles are all the same thing (called a comic). The only difference is if it includes or goes through the point at infinity. The relevant branch of mathematics is called projective geometry.

Smaug123 6y ago

Good point; this becomes more obvious if you imagine throwing the ball up and then immediately collapsing all the mass of the Earth into a single point at the centre. What path does the ball follow now? It's probably following a path we would more usually call an "orbit", and it sure looks a lot like an ellipse. Now just put the mass of the Earth back where it was, and notice that the ball hits the ground before it can trace out very much of its orbit.

est31 6y ago
Despite both being conic sections, cutting up an ellipse won't yield you parabolas. An ellipse has two focal points to which the sum of the distances is constant, while a parabola has a focal point and a directrix line to which the difference of the distances is constantly 0. Two different things.
- Koshkin 6y ago
  Isn’t a conic section made by cutting up, well, a cone?
- srtjstjsj 6y ago
  More symmetrically, Every comic section has one focus and one directrix (and a semi-major axis for scale), eccentricity is the ratio of distance. Ellipse has eccentricity less than 1, and a parabola has eccentricity equal to 1.
- pdkl95 6y ago
  Parabolas are just a special case of ellipses where one focus is infinitely far away. Vi Hart has a nice explanation: https://www.youtube.com/watch?v=v-pyuaThp-c https://www.youtube.com/watch?v=v-pyuaThp-c

dmurray 6y ago

A good point to make, but if one is going to be that pedantic one shouldn't call it "impossible". A parabola is possible with the right wind, air resistance, gravity from nearby mountains, etc, and in practice those have a larger impact on almost any suborbital projectile than the difference that turns the trajectory from a parabola to an elliptical arc.

saagarjha 6y ago

Speak for yourself. My balls reach escape velocity and then make corrections to assume e=1.

fouronnes3 6y ago

It's a parabola in a uniform gravity field, an ellipse in a circular gravity field coming from a point mass. 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.

centimeter 6y ago
> it's never an ellipse because the Earth is not a point mass. 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.
- libraryofbabel 6y ago
  > 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 https://en.m.wikipedia.org/wiki/Schiehallion_experiment It all depends on how pedantic one wants to be :)
  - perl4ever 6y ago
    The first time I heard of the concept, it wasn't actually that, it was how in the 60s, they had to correct for mascons in the moon while orbiting.
    - garmaine 6y ago
      There are very few stable orbits close to the lunar surface. Basically a couple of polar orbits with very specific parameters, and that's it. The rest get so perturbed by gravitational anomalies that they fall out of orbit after a few months or years--faster than low Earth orbit where there is atmospheric drag! https://www.lpi.usra.edu/lunar/documents/NTRS/collection2/NASA_TP_3394.pdf https://www.lpi.usra.edu/lunar/documents/NTRS/collection2/NA...

gpsx 6y ago

If you want to get technical, what about air resistance?

ummwhat 6y ago

Ellipse, parabola, hyperbola and circles are all the same thing (called a comic). The only difference is if it includes or goes through the point at infinity. The relevant branch of mathematics is called projective geometry.