# Paper airplane physics

I am working on a game involving flying and steering a paper airplane for WP7. I want the plane to fly just like how normal paper airplanes fly (see this game for an example http://armorgames.com/play/7598/flight) but I can't seem to find an equation for how paper airplanes fly.

Anyone have any experience with this? In my game now, it just follows the usual motion for an object in a vacuum, which makes for some flight, but it doesn't feel perfect, and traveling at a slight downward angle makes you lose speed, which isn't right.

Thoughts?

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What's giving you lift if you're using the equations for motion in a vacuum? – Jerry Schirmer May 16 '11 at 0:47
Welcome to Physics.SE. I find the question very unclear (including the part where you explain what you current model is). Would you care to elaborate? – dmckee May 16 '11 at 0:59
Heres some elaboration on what I am doing now. The paper airplane starts with a "flick" (the user flicking his finger accross the screen, which throws the plane). This gives it its initial velocity. It then flies through the air, with a constant drag force, and a constant gravity pushing it down and slowing it. I'd like it to behave more realistically, more specifically, respond to the angle of the plane. If you fly straight up, your forward velocity is going to slow dramatically, and if you point slightly down, you wouldn't lose speed at all. – Woody Zantzinger May 16 '11 at 2:29
Just noticed your comment. If it's going upward it should slow down, and as it slows down it should curve downward and then pick up speed. When it picks up more speed than the speed it is stable at it should curve upward again. This oscillation should damp out so it is descending just at that stable speed. – Mike Dunlavey Oct 25 '11 at 2:43

There is no simple equation for how a paper airplane flies like there is for a simple projectile because the airplane can interact with the air in complicated ways.

The physics of a paper airplane is described by Newton's laws of motion. These laws apply to both the airplane and the air it travels through. The plane is acted on by a constant gravitational force and by contact forces with the air, especially drag and lift.

The nature of the force between the air and the plane can be quite complicated, and requires an extremely detailed analysis for accurate simulation. For example, by constructing the plane slightly differently, you can make it fly faster, slower, further, curve left or right, or bob up and down.

The basic physical ideas are those of fluid dynamics and the basic equation involved is the Navier-Stokes equation. Modeling something like an airplane accurately is mostly the domain of expertise of aeronautical engineers.

To make a simple model for a game, you might want to start with a simple constant gravity force, a drag force proportional to the square of the velocity, and a lift force also proportional to the square of velocity (which comes from here), and then play around with the parameters until you find something pleasing to your eye.

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I think the simplest way to solve my problem is to use the angle the plane is pointing to adjust how it gains or looses speed. A downward angle would gain speed, an upward angle would lose speed, and then just fiddle with those constants. – Woody Zantzinger May 16 '11 at 2:36
@Woody Okay, good. – Mark Eichenlaub May 16 '11 at 2:58
@Woody: If the plane is properly balanced (as in my answer), higher speed causes higher upward lift on the wing, and higher downward lift on the tail, so overall a higher nose-up moment, causing it to nose up and lose speed. Then with lower speed both lifts decrease, the nose drops, and the speed increases. This is the slow-rate phugoid oscillation, which all airplanes do and when it is damped out characterizes stable flight. – Mike Dunlavey Oct 24 '11 at 16:24
@Woody: Some planes, instead of a horizontal stabilizer at the tail, have a canard (or both) whose job is to support the nose against its heaviness. Same idea: slow down - nose drops - speed increases - nose comes up. – Mike Dunlavey Oct 24 '11 at 16:46

Paper airplanes are like real airplanes in their basic physics. Some points:

They should be mildly nose-heavy. (The tail actually presses downward, to counteract the nose-heaviness.) If they are too nose-heavy, they will just arrow into the ground. If they are tail heavy, they will go up, and then slide backward. If they are neutral-balanced, they will go up and down with a scalloped motion. If they are mildly nose-heavy, they will be stable, because if they slow down, the nose will drop, which makes them go faster, thus more lift, which brings the nose back up.

The speed is determined by how much up-elevator you put on the back. If you put a lot of up-elevator, they will tend to turn up, which slows them down, so they will be stable at a slower speed. If you put neutral elevator, they will have to be going much faster to bring the nose up, so they will tend to fly faster.

A paper airplane, like any airplane, will always descend unless something is pushing it. That's because by descending it is using gravity to overcome its drag and keep its speed up. If you want it to stay up longer, trim the elevator up so that it travels more slowly. Also, anything you can do to reduce drag will help it stay up.

If you want it to go in a straight line, rather than turn, all you can do is try to balance it left-to-right. That's a problem with airplanes in general. There's very little you can do to make them stable in the roll axis. That's why when pilots wander into clouds, where they can't see a horizon, they can easily get into a spiral, unless they can keep the wings level by trusting their instruments.

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@Downvoter: I know it's informal, but did I say something incorrect? – Mike Dunlavey Oct 22 '11 at 0:28
I'm not your downvoter, but I imagine it was because you did little to really answer the question. This is vaguely general advice for building serviceable paper airplanes - it does little to explain or assist in the understanding of a mathematical flight model for paper airplanes in general. – J... Jul 2 '14 at 2:16
@J...: The OP had some basic misconceptions that needed to be corrected. When that is done, the equations just fall out. I was trying to correct the misconceptions. – Mike Dunlavey Jul 2 '14 at 13:32

It's really not that complicated!

I would model the energy flows between kinetic and potential energy. Start with a full bucket of potential energy plus the appropriate speed and drain kinetic energy away over time depending on flight speed.

To know how much energy is lost to drag, you need to model the drag using two components:

1. Friction drag, which grows with the square of airspeed, and
2. Induced drag, which drops with the square of airspeed.

If your drag is the sum of both, its minimum will be at some moderate speed. I plotted the drag components for a glider below, but since the physics are the same for a paper airplane, this plot should do for now.

The nonlinear behavior of the induced drag curve at low speed is due to flow separation, and something very similar will happen for a paper airplane. The important thing is: The drag curve has a minimum.

The energy loss over time is drag $D$ times speed $v$. This energy $E$ has to come from the reduction of height $h$ over time $t$: $$\frac{\delta E}{\delta t} = \frac{\delta(m\cdot g\cdot h)}{\delta t} = D\cdot v = m\cdot g\cdot v_z$$ with $m$ the mass of the paper airplane, $g$ gravitational acceleration and $v_z$ the vertical speed.

For picking a realistic drag it helps to rephrase the equation above by introducing lift $L = m\cdot g\cdot n_z$. $n_z$ is the load factor and is approximately one in straight flight. The ratio between lift and drag for a paper airplane is somewhere between 4 and 10 - just pick a number which results in a realistic simulation. To calculate the sink speed $v_z$ as a function of flight speed $v$ use this formula: $$v_z = \frac{c_{D0}\cdot S\cdot v^3}{m\cdot g\cdot\rho} + \frac{m\cdot g\cdot\rho}{v\cdot\pi\cdot b^2}$$ The first term on the right side comes form friction drag and the second term from induced drag. S is the wing area of the paper airplane and $b$ its wing span. $\rho$ is air density, and for the zero-lift drag coefficient $c_{D0}$ you should pick a number which makes the paper airplane look realistic. Start maybe with 0.05.

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