Teen solves Newton’s 300-year-old riddle

[thriftyandboringinohio]

Re: Teen solves Newton’s 300-year-old riddle

I'm not going to get into the equations, I have not done that since college. But riddle me this: what is new here? Are we concluding that this is a mathematical expression of known empirical results that has never been quantified in a short equation in this manner? I am going on a limb and saying this finding will not revolutionize applied ballistics in any way. Or please correct me if I am wrong.

Your are correct, cbr. It's just a clever bit of math to find a closed-form solution to an old puzzle. Nerds like me find it worthy of celebration.

[leegs]

Re: Teen solves Newton’s 300-year-old riddle

Just at a glance, that solution does not look correct to me.

I don't expect "m" in the expression for drag forces in the third line.
I presume "m" is mass, which plays no role in classic aerodynamic drag. That force depends only in the shape of the body, the velocity of flow, and the properties of the fluid (i.e. air).

The whole thing looks off the mark (though I admit to only spending 90 seconds looking at it, and its been many years since I've worked any problems in fluid dynamics or Newtonian motion).
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Nerding out here (and probably incorrectly) . . . Mass may not have a role in aerodynamic drag, but it does have a role in the trajectory of a projectile.

[Jam]

Re: Teen solves Newton’s 300-year-old riddle

I'm not going to get into the equations, I have not done that since college. But riddle me this: what is new here? Are we concluding that this is a mathematical expression of known empirical results that has never been quantified in a short equation in this manner?

There are various levels of understanding physical reality in mathematical terms. For some, (1) writing the Hamiltonian (or Lagrangian) of a physical system conveys the deepest understanding available. Then, there is (2) the equation of motion (often, a differential equation), and finally, (3) a solution to the equation. Engineers go a step further, and use that knowledge to design and build machines, which exploit the understanding. The equations of motion for a projectile have been known since the times of Newton. Their solution, apparently, have not been known until now (although there is a lively dispute about that going on the net). However, the solution provided by Ray is not exactly what one would expect when hearing the word 'solution.' When I heard it first, I thought it would be in the form (x(t),y(t)), where x(t) and y(t) are the coordinates of the projectile at time t, which obviously, it is not. Instead, it is in the form of an algebraic equation connecting the horizontal and vertical components of the velocity. The closed-form solution (x(t),y(t)) would still require some work, some of which would be numerical. Another way to put it is that the system of differential equations has been reduced to an algebraic equation. One might think that replacing one equation with another is not a big deal, but, generally, solving numerically an algebraic equation requires less computational resources than solving differential equations, the latter often referred to as 'doing computer simulations.' Yes, the differential equations governing the motion of a projectile have been solved (simulated) numerically to great precision. Also, with current computer hardware, and efficient algorithms, the simulations can be done very fast. Hence our ability to predict the motion of a projectile before we fire it. Providing an algebraic equation enables a different level of understanding of the physical system. In the case of the projectile motion discussed here, I would put that level between (2) and (3) as defined above.

I am going on a limb and saying this finding will not revolutionize applied ballistics in any way. Or please correct me if I am wrong.

You are probably right. But, you never know... Perhaps the computational efficiency enabled by the reduction of differential equations to algebraic equation will allow having a small dedicated chip built into a bullet that will control its flight so as to hit an incoming bullet if it determines that the incoming bullet would otherwise hit a protected asset.
And, if nothing else, the clever trick that Ray used in solving this equation may, perhaps, find use in solving other intractable equations.

[thriftyandboringinohio]

Re: Teen solves Newton’s 300-year-old riddle

There are various levels of understanding physical reality in mathematical terms. For some, (1) writing the Hamiltonian (or Lagrangian) of a physical system conveys the deepest understanding available. ...

That's an important little point. Numerical methods and empirical K factors can give very useful results, but don't reveal the more fundamental truths.