Law of Gravity Deduction from Parabolic Trajectory?

[Islam Hassan]

Could one deduce mathematically the inverse-square nature of the law of gravity from measurement of the parabolic trajectory of a given projectile in earthbound flight? Could one further deduce the entire law itself from the same?

IH

 

[vanhees71]

No, one couldn't. You'd have to measure the trajectory much more accurately since in reality as a bound motion it must be part of an ellipse.

One can, of course, deduce the Newtonian Gravitational Force Law from Kepler's Laws of planetary motion. That's in fact the way, Newton has found it! It's a somewhat funny story since Newton didn't like to publish such profound findings in physics. It has needed a very stubborn Halley to pursuade him to do so. Legend says that Halley once told Newton that he'd like to know the force law behind Kepler's Laws, and Newton just answered that he'd found the answer quite a time ago and then looked for it among a huge pile of papers containing complicated calculations. Only on long insistence of Haley's, he finally wrote one of the most important books ever written, namely the "Principia Naturalis", but there he didn't make use of his other great invention, the "mathematics of fluxions", which we nowadays call differential and integral calculus, but tried to derive everything from Euclidean geometry. That makes it not so easy for us nowadays to understand the Principia, and that's why famous modern physicists like Feynman or Chandrasekhar wrote books explaining the Principia for the modern physicist.

 

[Islam Hassan]

No, one couldn't. You'd have to measure the trajectory much more accurately since in reality as a bound motion it must be part of an ellipse.

Many thanks for the feedback; does that mean that any earthbound projectile trajectory is locally a parabola but somehow melds into a more general ellipse? What is the nature of this ellipse? What does it circumscribe?

IH

 

[mathman]

The general solution to the two body problem is a conic section of some sort depending on the initial conditions.

 

[Vanadium 50]

Can you elaborate on this?

An orbit in a general r^a potential is a rosette, unless a = -1 or 2, in which case you can have closed orbits. I don't see how you can slice a cone to get even a section (e.g. a fraction of a lobe) of a rosette.

Getting back to the original question, you need to measure the deviation from a parabolic orbit to infer the inverse square law. This is at the part per million level, so is hopeless. You're looking for a bacterium sized deflection of a baseball-sized object.

What you can do, though, is use something called an "absolute gravimeter", which calculates the gravitational acceleration by timing an object falling in vacuum. These have precisions sufficiently good that you could measure g on two different floors of a building and determine the inverse square nature of gravity. The commercial state of the art using this technique will in principle tell you the exponent must be somewhere between 1.9999 and 2.0001. In practice you will quickly learn that you are limited by not by the ability to measure accelerations but by your knowledge of the material distribution in the surrounding city.

To get away from this, people go to towers out in the middle of nowhere. I think the best measurements are still those of Jim Thomas et al, 20 years ago. If I did the math right, their limit corresponds to an exponent between about 1.97 and 2.01. One of their problems was the gravitational attraction of the atmosphere.

 

[mathman]

The standard two body problem has a = -1. Newton solved this problem himself.