# What are the flight formulas for aircraft

I am making a 2D flight "simulator" (trying to be as realistic as you can for 2d) and I need flight formulas for lift, speed and pitch. I have tried a few sites but I do not understand what they are saying. Can someone please tell me the answer (if this is needed i am using the Boeing 787-8 dreamline). Sorry if this post is not formatted/written properly this is my first time using this site.

• What you're looking for is termed "flight dynamics" and takes up an entire book, like this one for example. Out of curiosity, how are you going to solve the equations of motion? Jun 26, 2023 at 16:31
• Is there a simple formula? Jun 26, 2023 at 16:33
• Can you explain what a 2D flight simulator consists of? (i.e. which two dimensions?) I'm picturing an overhead view, Instrument/Navigation trainer with a moving map type display. If this is the case you should be able to greatly simplify the underlying code vs 3D. Jun 26, 2023 at 16:45
• Oh sorry i mean like side view not overhead. Jun 26, 2023 at 17:35

Your '2D' problem (altitude, downrange) is really a 3DOF problem, (adding a rotation -- pitch angle).

Normal 3D flight is a 6DOF problem -- 3 positions, 3 angles.

You will need to write the equations for the accelerations of each DOF and then integrate them twice to the velocities and positions. This is a coupled 2nd order ordinary differential equation. You will start at an initial position and time-step your way through the solution.

You need control variables (pilot inputs). These will normally be throttle and elevator angle. If the aircraft is a glider, then you only have elevator angle.

A big decision is whether you want to represent the aircraft as a point mass, or as a distributed mass. The point-mass assumption will set the pitching moment of inertia to zero. This will make the angular equation of motion respond instantly. You can then solve the pitch equilibrium equation as a statics equation (trim) or you can ignore it and treat either angle of attack or CL as your pitch input directly.

I would recommend you start with an even simpler problem. A point-mass with no thrust and no drag (and no lift). I.e. just mass. This is the cannonball problem. You can derive the equations of motion the same way and then implement an ODE solver to plot out the path of the cannonball given an initial Vx,Vy.

Once you have the cannonball working, I would add drag and then thrust, then lift. When you add thrust, you can add a throttle control. When you add lift, you will need to add the pitch control of some sort.

I would start with point-mass equations (ignore the pitch inertia) and only add it if when you get finished with everything else, you don't feel it is realistic enough.