math - How to find trajectory path if the distance traveled, starting elevation, ending elevation and time in the air are known

Question

I'm trying to find the trajectory an object would take (assuming air drag is negligible and trajectory angle is always between 0 and 90), including peak height. Here's what I know:

• takeoff point latitude, longitude, elevation
• landing poing latitude, longitude, elevation
• distance traveled
• amount of time in the air

For me the problem could be reduced to 2d space, so if an object travels a horizontal distance of 8 meters starting at an elevation of 0 and ending at a relative elevation of -5 meters, and travels for 2 seconds. What was the peak height, and at what point in time did it reach that peak hight.

Also, trying to write this equation in to php, C, or objective C. Thanks

Answer 1

I assume cannon ball like trajectory

• object has some start velocity
• not slowing by air (specified by question)
• only gravity force apply

So

1. solve the equations to find missing data

   

   g=9.81; // [m/s^2]
   t=!; // [s] time in air
   p0(x0,y0)=!; // [m] start point
   p1(x1,y1)=!; // [m] end point
   v0(vx0,vy0)=?; // [m/s] start velocity vector
   
   // Y axis free fall with initial velocity
   y1=y0+vy0*t-0.5*g*t*t
   vy0=(y1-y0+0.5*g*t*t)/t
   vy0=(y1-y0)/t+0.5*g*t
   
   // X axis constant speed
   vx0=(x1-x0)/t
   

2. trajectory path point

   x=x0+vx0*t;
   y=y0+vy0*t-0.5*g*t*t;
   

   where t = <0.0,time_in_air> so for plotting do some for loop through t

3. peak point:

   vy(t)=vy0-g*t;
   // for peak the y velocity is zero so
   0.0=vy0-g*t;
   t=vy0/g;
   

   where t is the time when object reach peak point. If you need coordinates then compute them like in bullet #2