This figure shows the various flight conditions encountered by an airplane.

The force system for straight and level flight. Lift = Weight.

At low speeds to fly straight and level, the airplane angle of attack is large whereas for high speeds the airplane angle of attack is small.

Figures (a) and (b) show the force systems for an airplane in a straight-constant velocity climb or dive an angle to the horizontal direction of flight.

This figure shows the force systems for an airplane in a straight, constant velocity vertical climb (straight up).

#### Performance – Class 1 Airplane Motions

Four basic forces act on an airplane: lift, drag, weight, and thrust. Additionally, in curved flight another force, the centrifugal force, is present. Performance is basically the effects that the application of these forces has on the flight path of the airplane. For purposes of examining performance, the airplane is assumed to possess stability and a workable control system.

Motions of an Airplane

All the motions made by an aircraft may be grouped into one of three classes: (1) unaccelerated linear flight—Class 1 motion, (2) accelerated and/or curved flight—Class 2 motion, and (3) hovering flight—Class 3 motion.

Class 1 Motion

Although straight and level flight may occur only over a small section of the total flight, it is very important since it is usually considered the standard condition in the design of an airplane.

For straight and level flight, the flight path is horizontal to the Earth's surface and, for simplicity, it is assumed that the thrust always acts along this horizontal plane. For the flight to be horizontal, or at a constant altitude, lift must equal weight. To fly at a constant velocity (unaccelerated), the thrust must equal the drag.

The velocity of the airplane must be sufficient to produce a lift equivalent to the weight, and there is a range of velocities over which the plane may fly straight and level. Thus, combining the equation for the coefficient of lift:

CL = L/qS

where L = Total lift on the wing, S = wing area, and q = dynamic pressure

with the condition that Lift = Weight, one obtains

 Weight = 1/2r¥V¥2CLS (1)

If it is assumed that the weight, air density r¥, and wing area S are constant, one observes that as the velocity V¥ increases, the wing lift coefficient CL decreases, which may be accomplished by a decrease in the wing angle of attack. Minimum flying speed for straight and level flight occurs when the wing is operating at CL,max, that is, near the stall angle. The maximum flying speed for straight and level flight is limited by the thrust available from the engine. This condition also requires a small value of CL and hence a small angle of attack.

Thus, at low speeds, to fly straight and level, the airplane angle of attack is large whereas for high speeds the airplane angle of attack is small.

For a straight, unaccelerated ascent (climb) or descent (dive), with the assumption that the thrust line lies along the free-stream direction or flight path, the climb or descent angle is given by +y or -y [lower case Greek letter gamma], respectively. If the forces are summed parallel and perpendicular to the flight path, it is seen that the weight force is resolved into two components. One obtains

 L = W cos y = W cos (-y ) (Climb or dive)
 T = D + W sin y (Climb)
 T = D + W sin (-y ) = D - W sin y (Dive) (1)

To maintain a straight climbing (or diving) flight path, the lift equals the component of weight perpendicular to the flight path.

L = W cos y = W cos (-y)

In the case of the climb condition to maintain a constant velocity, the thrust must equal the drag plus a weight component retarding the forward motion of the airplane. In the case of the dive condition, the weight component along the flight path helps the thrust by reducing the drag component for constant velocity.

The conclusion is that one must use an increased thrust to climb at constant velocity and use less thrust to dive at constant velocity. This is analogous to the situation of a car where one must "give it the gas" (apply more thrust) to prevent the car from slowing down in going up a hill and "let up on the gas" (use less thrust) to prevent the car from speeding up when going down a hill.

It is interesting also to examine three special cases of the use of the above equations. First, in straight and level flight, the climb angle y is zero, hence sin y = 0 and cos y = 1. This yields the previously derived conditions that Lift = Weight (L = W) and Thrust = Drag (T = D). Secondly in a vertical climb y = 90°, and hence sin y = 1 and cos y = 0. Thus, the thrust necessary to climb vertically is equal to the drag plus the airplane weight (T = D + W). Also, for a vertical climb, the lift equals zero (L = 0).

The final condition to be discussed is gliding flight. In gliding flight, the thrust equals zero. It is therefore necessary to balance the forces of lift and drag with the weight. Equation (2) remains unchanged but equation (1) is simplified. In a glide:

 L = W cosyg (2)
 D = W sinyg (3)

If one divides equation (2) by equation (3), the result is:

 L/D = 1/tanyg

In nonmathematical language, this means that the smallest glide angle, and hence maximum gliding range, is obtained when the lift-drag ratio is the maximum. The lift-drag ratio is a measure of the aerodynamic efficiency of the airplane. Sailplanes possess the greatest lift-drag ratios with excellent aerodynamic design since they rely on air currents to keep them aloft. For a particular airplane, the lift-drag ratio varies with the angle of attack of the airplane (not to be confused with the glide angle of the flight path). There is a particular angle of attack for which this ratio is a maximum. This is then the angle of attack for minimum glide angle and maximum range. For any other angle of attack, the lift-drag ratio is less and the glide angle is increased; hence, a steeper glide results. It is a natural tendency for a pilot to raise the airplane's nose (increase the angle of attack) to try to get maximum range, but unless this gives the maximum lift-drag ratio, the descent will be steeper instead.

—Adapted from Talay, Theodore A. Introduction to the Aerodynamics of Flight. SP-367, Scientific and Technical Information Office, National Aeronautics and Space Administration, Washington, D.C. 1975. Available at http://history.nasa.gov/SP-367/cover367.htm