flash - How to use a parabola formula in AS3 for firing an arrow that will always intercept a given point -

first note: mathematically, i'm not skilled @ all.

i played game on iphone while press point, , arrow fires castle intersect point pressed. wanted make similar game, thinking easy quick make; ran realization mathematics beyond skill level.

i'm assuming they're using parabola formula or determine velocity , angle needed when arrow launched arrow intersect clicked point.

i vaguely remember how parabolas work school , have no chance of working out formulas.

any mathematical or ideas might easier implement great.

i want end function in castle so:

package {     import avian.framework.objects.avelement;      public class castle extends avelement     {         /**          * fires arrow          * @param ix x intersection point          * @param iy y intersection point          */         public function fire(ix:number, iy:number):void         {             var ar:arrow = new arrow();              ar.x = x;             ar.y = y;              // define angle , velocity based on ix, iy             // ar.fireangle = ??             // ar.firevelocity = ??              parent.addchild(ar);         }     } } 

update per questions in comments:

there no forces applied arrow such wind, friction, etc. also, starting point of arrow fixed throughout game (at castle).

here example image more clarity:

to clear possible:

1. arrow begins journey fixed point (say: 40, 120).
2. the arrow must intercept given coordinate.
3. a realistic possible path i'd achieve (obviously can fire arrow straight intercept point, goal have arrow first rise, descend; passing through desired coordinate @ realistic point in journey).

note: avoid issue of there being infinite possible parabolas - velocity of arrow can fixed - @ defining angle arrow can leave at.

the flight path of projectile through gravitational field can described applying equations of motion

the equations use are

1. v = u + @ 2. s = ut + (at^2)/2 

where

s = distance between initial , final positions
u = initial velocity
v = final velocity
= constant acceleration
t = time taken move initial state final state

ok. animate arrow calculate new velocity , position @ regular intervals (every frame) based on previous velocity, position , acceleration. acceleration in case entirely due gravity.

lets simplify , measure time intervals in frames rather seconds. gives t = 1 above equations allowing rewrite them as

1. v = u + a*1           => v = u + 2. s = u*1 + (a*1^2)/2   => s = u + a/2 

now in x direction acceleration, = 0 (we're not taking drag account). in y direction = g, acceleration due gravity. if rewite these equations resolved each axis get

for x:

1. vx = ux + 0        => vx = ux (no change we'll ignore this) 2. sx = ux + 0/2      => sx = ux (convenient eh?) 

for y:

1. vy = uy + g 2. sy = uy + g/2  

so lets plug them sample script

public class arrow extends sprite {     //g constant     //it's closer 10 our world     public static const g:number = 2;      //our arrow     private var arrow:shape;      //start velocities     private var ux:number;     private var uy:number;      public function arrow()     {         arrow = new shape();         arrow.graphics.linestyle( 1, 0 );         arrow.graphics.lineto( 30, 0 );     }      public function fire( vx:number, vy:number ):void     {         ux = vx;         uy = vy;         addchild( arrow );         addeventlistener( event.enter_frame, fly );     }      private function fly( e:event ):void     {         //lets use our equations         var sx:number = ux;       //distance moved in x dir          var vy:number = uy + g    //new velocity in y dir         var sy:number = uy + g/2  //distance moved in y dir          //apply arrow         arrow.x += sx;         arrow.y += sy;          //save new y velocity         uy = vy;          //extra bonus rotation of arrow point in right direction         arrow.rotation = math.atan2( uy, ux ) * 180 / math.pi;      }  }