SeaGypsy wrote:Without seeing the data maybe hard to say, but likely similar to fluid dynamic equations also, meaning new solutions in all kinds of hull design. If this kid's formula really works, it could make designing an aircraft, a ship, a projectile, child's play. Just put in the approximations/ dimensions, variable mass etc. & out pops your ideal configuration. For this reason, plus the massive number of people working in industrial design who are likely to become redundant, it would not surprise me if the formula disappears into a vault somewhere.
It's not that useful..
First there is already a numerical solution, the kid only found the analytical solution. Two the drag is proportional to the velocity square. But in the real world, the drag is proportional to "a v + b v^2 + c v^3 + d v^4". So as you can see it's only an approximation to the real world..
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The problem he solved is as follows:
Let (x(t),y(t)) be the position of a particle at time t. Let g be the acceleration due to gravity and c the constant of friction. Solve the differential equation:
(x''(t)2 + (y''(t)+g)2 )1/2 = c*(x'(t)2 + y'(t)2 )
subject to the constraint that (x''(t),y''(t)+g) is always opposite in direction to (x'(t),y'(t)).
Finding the general solution to this differential equation will find the general solution for the path of a particle which has drag proportional to the square of the velocity (and opposite in direction). Here's an explanation how this differential equation encodes the motion of such a particle:
The square of the velocity is:
x'(t)2 + y'(t)2
The total acceleraton is:
( x''(t)2 + y''(t)2 )1/2
The acceleration due to gravity is g in the negative y direction.
Thus the drag (acceleration due only to friction) is:
( x''(t)2 + (y''(t)+g)2 )1/2
Thus path of such a particle satisfies the differential equation:
( x''(t)2 + (y''(t)+g)2 )1/2 = c*(x'(t)2 + y'(t)2 )
Of course, we also require the direction of the drag (x''(t),y''(t)+g) to be opposite to the direction of the velocity (x'(t),y'(t)). Once we find the intial position and velocity of the particle, uniqueness theorems tell us its path is uniquely determined.