# How to model the flight kinematics of an airplane with air resistance?

1. Aug 31, 2014

### Meowy

I want to derive some equations for a 2-D airplane's motion, specifically when it lifts off and lands. Can someone show me where I can start and how I can use differential equations to develop equations like the speed/position of the airplane in its x and y components?

Also, if anyone is wondering, the assumptions I'm making is that there's an air resistance force that acts opposite to the instantaneous direction of motion and is proportional to the speed of the aircraft at that point (also constant lift coefficients, pressure, area). I also want to assume that the aircraft is accelerating with a constant force.

Any suggestions or help is appreciated :)

2. Aug 31, 2014

### Simon Bridge

OK, a very simple model.
You want a constant thrust and given lift coef, and const. weight. Weight always points down.
Does the thrust always point in the same direction? What about lift?

You want to model drag as a force proportional to the velocity, in the opposite direction?

All you need are Newton's laws, and the ability to draw vectors and solve differential equations.

3. Sep 1, 2014

4. Sep 2, 2014

### Meowy

Thanks for the links and help! However, I'm a bit confused as to how the vertical and horizontal components of the differential equations associate with each other? For instance, I think the basic equation to integrate with for both the x and y components are dv/dt = a -kv^2 for each where a is like the acceleration of gravity for y component and the acceleration of the plane for x component (k being constant contributing to air resistance).

So, integrating each of the differential equations can get me an equation in terms of the instantaneous Vx or Vy of the plane and then going even further, the positions. But is there a way to somehow link the vx and vy and their respective equations?

Also, I'm wondering how it's possible to model a motion if the plane's acceleration or lift is starting at an angle? Can I, for instance, say acos(θ) is the horizontal component of that acceleration and use that in the equation?

5. Sep 3, 2014

### Simon Bridge

If your drag depends only to first order on velocity, the horizontal and vertical components don't mix.

Calling the drag force D, then:
$\vec D= -k\vec v$ then $(D_x,D_y) = -k(v_x, v_y)$ where k is the drag coefficient.

But if you have $\vec D = -k\vec v^2$ then that is more of a problem - the x and y cmponents of velocity do mix. You will have to multiply out the terms to get a pair of couple DEs.

Considering drag D, thrust T, weight Mg, and lift L(v), you need to do:
$$M\frac{d}{dt}\vec v = \vec T +\vec D +\vec L +\vec L$$ ... plug in your models for each force.

You'll have to do this as an initial value problem - so solve it for $x(0)=x_0, y_0=y_0, \vec v(0)=v(\cos\theta, \sin\theta)$. That takes care of the initial angle thing.

Note: in general the equations governing flight can get arbitrarily complicated and are usually not solved algebraically. You would need to impliment some sort of dynamical calculation for small time-steps or something.