The total takeoff distance needed consists of three parts: (1) the ground-roll distance, (2) the transition distance, and (3) the climbout distance over an obstacle.
Forces acting during take-off ground roll.
This figure presents the forces acting on an airplane during the landing rollout.
The disposition of forces in a properly executed turn. Vertical lift = weight.
Hovering flight. Thrust = Weight.
Vertical ascent and descent.
Early VTOL airplanes.
Performance - Class 2 and Class 3 Motion
Accelerated motion and curved flight or Class 2 motion, is considered specifically for the cases of takeoff, landing, and the constant-altitude banked turn.
The takeoff of an airplane is a case of accelerated motion. From the instant the airplane begins its takeoff roll, to the time it begins its climbout after leaving the ground, it is under continuous acceleration. The total takeoff distance needed consists of three parts: (1) the ground-roll distance, (2) the transition distance, and (3) the climbout distance over an obstacle.
During the ground roll, in addition to thrust, weight, drag, and lift, there is a rolling frictional force due to the landing gear. The sum of the forces in a horizontal direction is equal to the net force acting to accelerate the airplane down the runway. At the beginning of the ground roll, lift and drag are zero as dynamic pressure is still zero (assuming no winds). Acting under the net acceleration (thrust exceeding the total retarding force), the velocity increases and lift and drag build. The airplane remains in a horizontal attitude until some velocity (about 10 percent above the airplane stall velocity for safety) is reached at which point the airplane is "rotated" or pitched up. The pitch increases the airplane angle of attack, the lift quickly exceeds the weight, and the airplane leaves the ground. Rolling friction forces drop to zero at liftoff, and the airplane's total drag decreases greatly as the landing gear is retracted. At the end of transition, about 20 percent above the stall velocity, the airplane begins its climbout, usually at constant velocity. The ordinary equations for climb apply in this case.
The total distance for the airplane to clear from the start of its roll is important and determines the amount of runway required for design purposes. Additionally, the pilot should know the maximum speed from which the takeoff may be aborted so that sufficient runway exists for deceleration to a stop.
The takeoff distance may be reduced by the use of flaps and other high-lift devices. However, there is a limit to their use since they also contribute to increased drag and retard the airplane's acceleration. There is usually an optimum flap setting for an airplane that will minimize the takeoff distance. Some airplanes may also use rocket-assisted units to take off in the minimum distance. These units represent a transitory increase in thrust and provide a means of high acceleration for short periods. On board an aircraft carrier, this method takes the form of a catapult, where flying speed is achieved in a matter of a second or two.
Landing an airplane consists of touching down at the lowest possible vertical and horizontal velocities and ground rollout.
Under touchdown conditions, it is assumed that the vertical velocity is near zero and that the lift equals the weight. Flaps are used advantageously to decrease the landing velocity. Indeed, they increase the maximum lift coefficient and decrease the landing velocity as indicated by:
During the landing rollout, the forces acting on an airplane are the same as during takeoff except for their magnitude and direction. The rolling friction is greater as the brakes are applied. For safe operation, this condition occurs near the end of the rollout. Spoilers on the wings are used to "dump" the airplane lift to prevent the airplane from rebounding into the air after touchdown. This condition increases the rolling friction as the normal (perpendicular) force is increased. The engine thrust is zero or, more usually for large commercial and military airplanes, is negative. This condition is accomplished by using reversible pitch propellers or thrust reversers. For ground roll during landing, the thrust force is retarding. The airplane drag may be increased by setting the flaps for maximum drag. Therefore, there is a net deceleration acting on the airplane to slow it to a stop. Another favorite braking device used by military airplanes is the parachute that is opened at touchdown. On board aircraft carriers, the usual landing brake is mechanical in the form of the arresting hook on the airplane engaging a cable laid across the flight deck. Deceleration is exceedingly swift and the airplane is subjected to large structural forces.
Not all motions of an airplane are in a straight line. There are ample cases of curved flight paths. These cases include the climbing and descending turns, maneuvers in combat, and aerobatics. One of the basic maneuvers required to change the flight-path heading is the constant altitude banked turn.
In a turn, accelerations due to a change of direction acquire added significance. By Newton's first law, a body in motion in a straight line will continue in motion in that same line unless acted upon by an external force. To maintain an airplane in a curved path requires that an acceleration be supplied toward the center of the curve. By Newton's second law, the force required to perform this, called centripetal force, is proportional to the acceleration required to maintain the curved flight. By Newton's third law, there is a reactive force by the body, opposite the centripetal force, called the centrifugal force. The centrifugal force is given by:
FC = mV¥2/R
where m is the mass of the airplane, V¥ is the velocity of the airplane in the curve, and R is the radius of the turn or curved flight path. From this equation one sees that the highest centrifugal forces occur for massive airplanes at high speeds in tight turns.
In a properly executed turn, the wings are banked at an angle ø [Greek letter theta] to the horizontal. This angle causes the resultant lift on the wings to bank also. When resolved into vertical and horizontal components, it is the horizontal component of lift that is the centripetal force needed to maintain the curved flight path. This force is balanced by the reaction centrifugal force. For a constant-altitude turn, the vertical component of lift must equal the weight. Thus, the total lift must be increased to maintain constant altitude when entering a banked turn.
The smaller the turning radius is, or the greater the velocity in a turn, the larger the banking angle must be. This is required to produce a large enough horizontal lift component to hold the airplane in the turn.
Class 3 motion has been assigned to a special flight condition; that of hovering flight. In hovering flight, there is no motion of the aircraft with respect to the atmosphere. As such, this results in no aerodynamic reaction forces of the aircraft on the whole, that is, no lift and drag forces. In equilibrium, the remaining forces, thrust, and weight, must be balanced. Hence, for hovering flight,
Thrust = Weight
By properly controlling the thrust, the aircraft may be made to rise and descend vertically. The chief advantage of such aircraft is their ability to land and take-off in small spaces without the use of long runways. Since they land and takeoff vertically they are called vertical takeoff and landing (VTOL) aircraft. They have the added distinction of being able to perform at high speeds as a conventional airplane in flight. This is why helicopters, although capable of hovering flight, are usually not included in this grouping. They are, at present, incapable of the speeds and maneuvers of conventional airplanes.
The first concepts to be tried were three "tail sitting" airplanes, the Lockheed XFV-1, the Convair XFY-1, and the Ryan X-13 Vertijet. The Lockheed and Convair used turboprop-powered contrarotating propellers to supply the vertical thrust needed whereas the X-13 was jet powered. The main problems with these VTOL airplanes were the tricky piloting maneuvering required in the takeoff and landing and the need to tilt the entire aircraft over into conventional flight. The next concept tried was to keep the main body of the aircraft in a conventional sense, but tilt the wing and engines from the vertical to the horizontal. The LTV-Hiller-Ryan XC-142A was such an aircraft.
Another concept was to use separate powerplants for vertical takeoff and landing and conventional level flight. But this added dead weight to each flight regime. For simplicity and efficiency, the Hawker Siddeley Harrier has been one of the best VTOL aircraft. This plane uses the concept of "vectored thrust" where four rotating exhaust nozzles deflect the exhaust from vertically down to directly behind. Control at low flight velocities and in hovering flight is supplied by reaction jets in the wing tips, nose, and tail.
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
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