Trajectory Calculation

In case there is no Drag Force

Velocity

V = V_{0}+at

Position

P = P_{0} + V_{0}t + \frac{1}{2}at^{2}

No Acceleration With Drag Force

Velocity

If Drag Value == 1, it means Velocity is reduced 1/24 at every each frame at 24 fps
Velocity at time can be calculated (1-1/24)^f (f = frame)

a\left (1-\frac{D}{fps} \right )^{F}

Position

Here, I put

for temporary.

Position will be a trajectory of . so it can be expressed by

\sum_{i=1}^{F}aDr^{i}

then that formula is

\sum_{i=1}^{F}aDr^{i} = a\frac{1-Dr^{F}}{1-Dr}

Wrangle Examples

Trajectory by Initial Velocity

vector g = {0,-9.8,0};

@P += @v*(Time) + g*pow(Time,2)/2;

@v = @v + g*Time;

Initial Vel and Drag(no force)

float step = Frame - 1;

float dragRate = 1 - TimeInc;

float dragingInTime = ( pow(dragRate,step) - 1 )/(dragRate - 1);

@P = @v *TimeInc * dragingInTime;

@v *= dragRate;

Initial Vel and Accel with Drag

vector g = {0,-9.8,0};

float step = Frame - 1;

float second = step*TimeInc;

float dragRate = 1 - TimeInc;

float dragingInTime = ( pow(dragRate,step) - 1 ) / (dragRate - 1);

vector constantPos = @v*TimeInc * dragingInTime;

vector accelPos = g*pow(TimeInc,2) * ( (step-dragingInTime) / (1-dragRate) );

@P += constantPos + accelPos;

@v = @v*pow(dragRate,step) + g*TimeInc*dragingInTime;

Target Trajectory and Time

float t1 = @trajectoryTime; //time to reach goal

vector S1 = point(@OpInput2,"P",@ptnum) - @P ;//target Position

vector g = {0,-9.8,0};

vector v0 = S1/t1 - g*t1/2;

@P += v0*Time + g*pow(Time,2)/2;

@v = v0 + g*Time;

Wikipedia