redbull stratos fall simulation

fall.js

// redbull stratos fall simulation 
// node fall.js > fall.json
//
// port to js from python by @pybonacci
// more info: http://pybonacci.wordpress.com/2012/10/15/el-salto-de-felix-baumgartner-en-python/ 
//

// >> [ std_atm.alt2density(h, alt_units='m', density_units='kg/m**3') for h in xrange(0, 43000, 1000)]
// air density each 1000 meters 
var AIR_DENSITY = [1.225, 1.1116425767380347, 1.0064902527218418, 0.9091220810460088, 0.8191294202172186, 0.7361158590983954, 0.6596971404886504, 0.5895010832794331, 0.5251675031230921, 0.4663481315454516, 0.41270653342889785, 0.3639180227868754, 0.3108281589022067, 0.2654832635841079, 0.22675346883692057, 0.19367373647373748, 0.16541981206327336, 0.14128768681425868, 0.1206760556449457, 0.10307133434187116, 0.08803486247736621, 0.07487366111494297, 0.06372734460638993, 0.05428027105802167, 0.046267360336640966, 0.039465817649795336, 0.03368824062938579, 0.02877687388504317, 0.024598816068418232, 0.021042018277291016, 0.018011940446444, 0.01542875529531276, 0.013225008308169557, 0.011261986276505282, 0.0096088887240681, 0.00821392101750628, 0.007034412466278842, 0.0060351284310494364, 0.005186909284991773, 0.004465570167211259, 0.0038510093666613036, 0.0033264840369233825, 0.0028780204548397533];

var ALTITUDE = 39000;
var h = ALTITUDE;
var time = 0;
var velocity = 0;
var g = 9.81;
var C_D = 0.4;
var A = 1.0;  // m^2
var m = 80 + 20;  // kg: body + backpack


// return air density for altitude
function rho(h) {
  var i0 = Math.floor(h/1000);
  var i1 = i0 + 1;
  var r0 = AIR_DENSITY[i0];
  var r1 = AIR_DENSITY[i1];
  var t = (h - i0*1000)/1000;
  var d = (1 -t)*r0 + t*r1;
  return d;
}

var data = [];

function step(dt) {
  velocity += dt*(-g + rho(h) * velocity * velocity * C_D * A / (2 * m));
  h += velocity*dt;
  data.push({
    altitude: h,
    velocity: velocity,
    time: time
  });
  time += dt;
}

while (h>0) {
  step(0.5);
}
console.log(JSON.stringify(data));

Ufff. This is not taking in consideration that the air density changes at different altitudes, and that at a certain altitude he opened his parachute, and therefore the acceleration and speed is not constant. That must be hard, but we have some control points. We know that at certain altitudes the speed was some specific, for example when he broke the speed of sound, and when he opened the parachute. Those could be used to create the function.

For example: "If calculations prove to be accurate, and Felix is successful in his attempts to control his position, he will accelerate from standstill to the speed of sound - that's 0 to approximately 690 miles per hour in 40 seconds or less."

Taken from http://www.redbullstratos.com/science/speed-of-sound/