37 Takeoff & Landing Performance

Introduction

The ability for airplanes to operate safely from airports requires that they be suitably designed to exhibit good takeoff and landing distances, plus some safety margins. The principal interest is in the runway length and overall distance needed for the airplane (at a given weight and density altitude) to take off from the runway surface and clear some obstacle (e.g., buildings or terrain) from the initiation of the takeoff roll. It must also be ensured that the takeoff speeds are low enough that in an emergency, the pilot can abort the takeoff and still have sufficient runway length available to decelerate to a full stop.

Photo of an airliner just after takeoff in its initial climb out with its landing gear still extended. The first stages of the climb profile will involve flying at the best angle of climb to clear any ground obstacles, and then building up sufficient airspeed to climb at a better rate to reach the desired cruise altitude.

The takeoff distance and speed at which the airplane can safely fly can be significantly reduced by using flaps and other high lift devices such as slats, an optimum flap/slat setting being configured to minimize the takeoff distance and maximize the initial rate of climb. For multi-engine airplanes, adequate single-engine out (one engine inoperative or OEI) performance is also essential. This requirement is because the loss of an engine at a critical point in the takeoff run may require the airplane to either stop in the remaining runway length or to continue the takeoff and climb on the remaining engine(s). For a landing, the airplane must also be able to touch down on the runway at the lowest possible airspeed and stop well within the available runway length. To this end, flaps, spoilers, reverse thrust (if available), and good wheel brakes are all important.

Objectives of this Lesson

  • Understand the factors affecting the takeoff performance of an airplane.
  • Understand the factors that can affect landing and stopping distances.
  • Be able to use appropriate equations to estimate the takeoff and landing distances of an airplane.

Takeoff Performance

During a takeoff maneuver, the airplane is in accelerated motion and experiences a continuously increasing airspeed. Finally, sufficient airspeed will be reached for the airplane to fly after covering a certain runway length. It then transitions into flight and climbs away from the ground, as illustrated in the figure below. The takeoff maneuver is relatively routine, but pilots always need to be prepared for unexpected emergencies, such as the loss of an engine or other mechanical malfunctions.

The takeoff distance is measured in terms of a ground-roll, some short transition distance, and then some additional distance needed to pass over a FAA-mandated 50-ft high obstacle.

The total takeoff distance for the airplane at a given weight and operating density altitude is determined from the sum of three parts:

  1. The ground-roll distance to the point where the airplane reaches a safe airspeed to be able to fly.
  2. The transition distance, where the pilot lifts the airplane’s nose to a climbing attitude, the lift builds up on the wing, and the airplane leaves the runway.
  3. The extra distance for the airplane needed to climb and pass over a fictitious 50-ft (15-m) high obstacle.

The fictitious 50-ft (15-m) high obstacle requirement recognizes that most runways inevitably have trees or some structure at their ends. FAA certification of airplanes per the FAA regulations (FARs) requires that the reported takeoff distances factor in this extra margin of safety to prevent pilots from underestimating the needed runway lengths – see CFR §23.2115 and CFR §25.105 and onward.

Takeoff Analysis

Estimates of the ground roll and takeoff distances for an airplane can be determined by using the basic principles of dynamics and aerodynamics. However, the solution of the resulting equations will require numerical integration, not least because the exact value of propulsive thrust is not necessarily known a priori; this latter issue also depends on whether it is a jet or a propeller airplane. Other factors, such as the rolling friction of the wheels on the runway, may not be known precisely, although estimates can be made for different runway surfaces, including both dry and wet. Nevertheless, reasonable estimates of the takeoff distance(s) are possible with some assumptions, which can help engineers to evaluate the various factors that will influence the takeoff performance of a new airplane design.

Consider an airplane of mass M (=W/g) and acted upon by a thrust T from the propulsion system. As shown in the figure below, there will also be a rolling resistance force which depends on the coefficient of rolling resistance \mu_r from the wheels on the runway surface.

The forces acting on an airplane during the takeoff roll. Rolling friction depends on the nature of the runway surface, and whether it is dry or wet.

At the beginning of the ground roll, lift and drag are zero because the dynamic pressure is still zero (assuming no winds). Rolling friction forces progressively decrease as the lift develops on the wings and becomes zero at the point of liftoff. As soon as the landing gear is retracted, the airplane’s drag decreases, and the airplane establishes its initial climb out.

The amount of rolling resistance R depends on the net vertical force on the wheels, which is given by

(1)   \begin{equation*} R = \mu_r ( W - L) \end{equation*}

where L is the lift produced and W is the aircraft’s weight. For a dry, paved runway, the coefficient of friction, \mu_r, is about 0.02. For a wet runway, this value may increase to as much as 0.1, so both engineers and pilots need to be aware that a wet runway can significantly increase takeoff distances. The lift on the wing is given by the standard aerodynamic formula as

(2)   \begin{equation*} L = \frac{1}{2} \rho_{\infty} V_{\infty}^2 S C_L \end{equation*}

There is also an increasing drag on the airplane as it builds up airspeed, which can be written in the standard way as the sum of non-lifting and lifting components, i.e.,

(3)   \begin{equation*} D = \frac{1}{2} \rho_{\infty} V_{\infty}^2 S C_D = \frac{1}{2} \rho_{\infty} V_{\infty}^2 S \left( C_{D_{0}} + \phi \frac{{C_L}^2}{\pi A\!R e} \right) \end{equation*}

The difference, in this case, is that the factor \phi is used to account for the proximity of the runway on the induced drag of the airplane, i.e., the so-called “ground effect,” which is explained later in this lesson in more detail.

While various equations have been used to represent this ground effect or “\phi factor,” one common approximation is to use

(4)   \begin{equation*} \phi = \frac{16(h/b)^2}{1 + (16h/b)^2} \end{equation*}

where h is the height of the wing above the runway and b is the wing span. Clearly, the phenomenon of ground effect is important near the runway (where \phi may be between 0.5 and 0.7 depending on the airplane), but \phi increases quickly to unity as h increases to one or more wing spans above the runway.

Therefore, the net (accelerating) force on the airplane will be

(5)   \begin{equation*} T - R - D = T - \mu_r ( W - L) - D = F \end{equation*}

where T is the thrust, R is the rolling resistance, and D is the drag. Because during the takeoff roll, the thrust exceeds the total friction and drag forces, then the airspeed increases. The resulting acceleration of the airplane, a, will be

(6)   \begin{equation*} a = \frac{dV_{\infty}}{dt} = \frac{F}{M} = \frac{ F g}{W} = \frac{(T - \mu_r ( W - L) - D) g}{W} \end{equation*}

so after a time t from the start of the takeoff roll, the airspeed will be

(7)   \begin{equation*} V_{\infty} = \int_0^t dV_{\infty} = \int_0^t \frac{F}{W} dt \end{equation*}

where F depends on V_{\infty}.

Equation 7 is the first integration required to estimate the takeoff distance. A second integration, therefore, gives the distance traveled during the acceleration to the liftoff (LO) point, i.e.,

(8)   \begin{equation*} s_{\rm LO} = \int_0^t V_{\infty} dt \end{equation*}

which can be used to help solve for the takeoff run if the airspeed at which the airplane will fly is known (or assumed). Alternatively, if the maximum lift coefficient of the wing is known (in its takeoff configuration), then the corresponding takeoff airspeed can be determined.

There are no closed-form solutions to the preceding equations (Eq. 7 and 8). Still, they do lay down the mathematical modeling principles that can be used to numerically solve the takeoff distance problem. Airplane manufacturers carefully do these calculations for different takeoff weights and atmospheric conditions (i.e., different density altitudes) and then verify the results using actual flight tests with the airplane. Representative results are shown in the figure below, obtained by numerical integration and using optimistic but realistic assumptions. The calculations can be repeated for cases of single-engine failure (with multi-engine aircraft) before the point of takeoff, i.e., by reducing the available thrust by half for a twin-engine aircraft. In this case, which is called a rejected takeoff, the required stopping distance on the remaining runway must be calculated.

Representative results for the solution to the equations of motion of a takeoff maneuver.

Takeoff Charts

At the end of performance flight testing, the results of the airplane’s takeoff performance are generalized for standard and non-standard ISA conditions and included in the flight performance charts for use by the pilots. The figure below shows a simple example for a general aviation airplane. These charts are mainly used for flight planning.

An example of a takeoff distance chart for an airplane, as would be included the operating manual.

From knowledge of the airplane’s weight, the prevailing values of pressure altitude (as measured on the altimeter), outside air temperature, and the winds, the pilot can use this type of chart to estimate the airplane’s anticipated takeoff distance. Remember that the basis of this chart is modeling and engineering data verified by testing. Notice again the significant effect of airplane weight on the takeoff distance, as well as increasing temperature above ISA standard conditions, i.e., the impact of increasing density altitude. Notice that a headwind can notably reduce the takeoff distances; the chart also suggests that taking off with a tailwind is never a good idea.

Some solutions to the equations for the takeoff distance are possible with certain assumptions and using certain approximations. Examples include constant thrust from the propulsion system, which is a reasonable assumption for a jet during the takeoff roll. One approximation for the takeoff distance, in this c ase, the “lift-off” distance s_{\rm LO}, is given by

(9)   \begin{equation*} s_{\rm LO} = \frac{1.44 W^2}{g \rho_{\infty} S C_{L_{\rm max}} ( T - [D + \mu_r (W - L)]_{\rm av} )} \end{equation*}

where the average resistive force from the drag and rolling friction is given by

(10)   \begin{equation*} [D + \mu_r (W - L)]_{\rm av} = [D + \mu_r (W - L)]_{0.7 V_{\rm LO}} \end{equation*}

The subscript 0.7 V_{\rm LO} means that this quantity is evaluated at 70% of the liftoff airspeed, which can be determined in terms of the estimated stall airspeed by using

(11)   \begin{equation*} V_{\rm LO} = 1.2 V_{\rm stall} = 1.2 \sqrt{ \frac{2 W}{\rho_{\infty} S C_{L_{\rm max}}}} \end{equation*}

with the factor 1.2 being included to give some margin for safety. Of course, this analysis is predicated on a knowledge of C_{L_{\rm max}} for the wing, a complication being that the wing operates in ground effect, as previously discussed, and so this value can only be estimated.

Further simplifications can be made, such as the resistive force is much less than the thrust force (again, this is reasonable for a high-performance airplane) to get

(12)   \begin{equation*} s_{\rm LO} = \frac{1.44 W^2}{g \rho_{\infty} S C_{L_{\rm max}} T} \end{equation*}

which gives a better basis to establish the primary effects that can be expected to affect the takeoff distance.

In summary, the preceding results show the following outcomes:

  1. The liftoff distance increases rapidly with the square of the airplane’s weight, meaning that if the weight is increased by 50%, then the runway needed will increase by over twice.
  2. The liftoff distance increases with decreasing air density (i.e., density altitude), explaining why airplanes take more runway length on hot days or when operating out of airports at high elevations.
  3. The liftoff distance is reduced by using more wing area and/or by increasing C_{L_{\rm max}}, which is why some flaps are used during takeoff but not so much as to increase the drag substantially.
  4. When the runway length is really short, military airplanes have been known to use rocket devices to increase net thrust. Onboard an aircraft carrier, catapults are used where flying speed must be attained in a matter of seconds.

Landing Performance

As they say, what goes up must come down, so eventually, the airplane needs to come in for a landing. The landing maneuver is typically performed at or near the lowest possible airspeed, keeping the landing distance as short as possible. This situation is where the application of wing flaps and slats (if the airplane has them) is advantageous because their use can increase the wing’s maximum lift coefficient by up to a factor of approximately two.

Analysis

The figure below shows the forces acting on an airplane during the landing rollout; they are the same as during the takeoff, except that the thrust will now be zero or negative (i.e., the use of reverse thrust, if available). The drag coefficient on the wing will be increased dramatically by fully deflecting the flaps downward and using spoilers. The rolling friction is much greater if the brakes are applied, with \mu_r = 0.4 being a typical value for braking on a dry paved runway. Brakes may only be applied for part of the landing distance, however, depending on the pilot’s discretion and the available runway length. Spoilers on the wings are used to decrease the lift on the wings and increase the weight onto the wheels to increase the rolling friction and braking action during the landing roll.

The forces acting on an airplane during the landing roll.

When the airplane touches down, engine thrust is brought to idle by the pilot, or with most jet and turboprop airplanes, negative or “reverse” thrust can be produced using thrust reversers or reverse (i.e., negative) pitch propellers. Because of the high landing speeds of many military airplanes, some may need to use a parachute deployed during the landing run. On aircraft carriers, airplanes are brought to a quick stop by engaging an arrester cable laid across the flight deck. However, this does impose large structural loads on the airplane as well as high decelerations on the pilot.

The same equations as for the takeoff apply here, but now T = 0 if not to a negative value, i.e.,

(13)   \begin{equation*} a = \frac{dV_{\infty}}{dt} = \frac{(-\mu_r ( W - L) - D) g}{W} \end{equation*}

the net negative force confirming a deceleration. If the lift on the wings can be dumped by using wing spoilers, then it is possible to assume L = 0 (or perhaps just a significantly smaller value than W) and so the weight of the airplane is entirely on its wheels and the deceleration will be

(14)   \begin{equation*} a = \frac{dV_{\infty}}{dt} = \frac{(- \mu_r W - D) g}{W} = \frac{ F g}{W} \end{equation*}

The corresponding reduction in airspeed can be calculated using

(15)   \begin{equation*} V_{\infty} = \int_0^t dV_{\infty} = \int_0^t \frac{F}{W} dt \end{equation*}

where t = 0 occurs at the touchdown with the second integration giving the actual landing distance, i.e.,

(16)   \begin{equation*} s_{\rm L} = \int_0^t V_{\infty} dt \end{equation*}

Landing Charts

Again, the preceding integrations would be performed numerically (with certain assumptions), and the results for the estimated landing distances would be verified by flight testing at different airplane weights and different atmospheric conditions. Along with the takeoff distance, the results will eventually be included in the flight performance charts for the airplane for use by the pilots; a simple example is shown below. Again, this type of chart will be used mainly for flight planning purposes.

An example of a landing distance chart for an airplane, which would be included the flight manual.

As with the takeoff problem, simplified solutions for landing distance are possible with certain assumptions and approximations. One result is that

(17)   \begin{equation*} s_{\rm L} = \frac{1.69 W^2}{g \rho_{\infty} S C_{L_{\rm max}} [D + \mu_r (W - L)]_{0.7 V_{\rm T}}} \end{equation*}

where

(18)   \begin{equation*} V_{\rm T} = 1.3 V_{\rm stall} = 1.3 \sqrt{ \frac{2 W}{\rho_{\infty} S C_{L_{\rm max}}}} \end{equation*}

with the factor 1.3 being included in this case to give some margin for safety.

Notice that the landing distances depend on the square of the airplane’s weight so that a heavier airplane will require much more runway length. Distances will also increase at lower air densities, i.e., higher density altitudes. Again, the importance of high lift devices becomes apparent, which will increase the C_{L_{\rm max}} of the wing in the landing configuration to reduce both landing speeds and required runway distances.
Remember that the previous methods and results are mathematical approximations, and better estimates of the takeoff and landing distances will require numerical solutions. Airplane manufacturers always do many flight tests to verify these numerical predictions before including the final takeoff and landing performance results in the flight manuals.

Effects of Flaps and Slats

From the preceding, it is clear that both the takeoff and landing distances are affected by the maximum lift coefficients attainable by the wing, i.e., by the value of C_{L_{\rm max}}, as shown in the figure below. This reason is why flaps and slats are used on airplanes, flaps alone being more common on low-performance airplanes and both flaps and slats being used on higher-performance airplanes such as airliners. Flaps can also be used to increase the wing area if they are designed to move aft as well as deflecting downward. Both flaps and slats are mechanical devices that are operated by the pilot as appropriate to the flight condition.

The maximum attainable lift coefficient depends on whether the flaps or slats are retracted or deployed. If the flaps and slats are retracted, then the wing is said to be in its “clean” configuration.

Slats, however, are usually found only on higher-performance and heavier airplanes. A slat is like a small secondary airfoil at the wing’s leading edge. When the slats are deployed, air flows through the gap and over the top surface of the airfoil, and the resulting flow helps to energize the boundary layer and delay the onset of flow separation and stall. The wing can then be flown at a higher angle of attack before stall with a commensurate increase in C_{L_{\rm max}}. The reduction in takeoff and landing distances with high-lift devices is significant.

Physics of “Ground Effect”

As previously mentioned, the ground effect is important in establishing takeoff and landing distances for an airplane. The physics of why this occurs requires further explanation. When an airplane nears the ground, the three-dimensional flow pattern around its wings is changed, primarily affecting the wing tip vortex formation and their positions relative to the wing. The consequence of this behavior is to alter the downwash field over the wing and its induced drag, as illustrated in the figure below.

The physics of “ground effect” manifest primarily as a decrease in downwash over the wing and so a decreased in the induced drag whenever the wing is within about one wing span of the ground.

The primary consequence of the “ground effect” is a decrease in the induced component of drag whenever the wing is within about one wing span of the ground, although some increase in lift is also produced. The consequence is that in ground effect operation, the wing appears aerodynamically to have a higher aspect ratio. Ground effect is often particularly noticeable to a pilot during landing, where in the landing flare, the airplane may appear to “float” just above the runway for some distance before the wheels touch down. While there may be some alteration in the aerodynamic characteristics of the fuselage and the tail in ground effect, these effects are usually minor compared to the effects on the wing. The results below, which were obtained with wings in wind tunnels, suggest that for more than one wing span above the ground, then “ground effect” can be considered as negligible.

Measurements of the reduction in drag on a wing operating in ground effect allows for a semi-empirical equation (curve-fit) to be obtained to aid in predictions of landing distance.

As previously mentioned, the induced drag on the airplane in ground effect can be approximated as

(19)   \begin{equation*} D_i = \frac{1}{2} \rho_{\infty} V_{\infty}^2 S \left( \phi \frac{{C_L}^2}{\pi A\!R e} \right) \end{equation*}

where \phi is given by the formula

(20)   \begin{equation*} \phi = \frac{16(h/b)^2}{1 + (16h/b)^2} \end{equation*}

and where h/b is the height to span ratio of the wing above the runway. The effective aspect ratio of the wing increases in ground effect, and is given by

(21)   \begin{equation*} A\!R_{\rm eff} = \frac{A\!R}{\phi} \end{equation*}

Notice again that the effects are only significant when the wing operates within one wing span of the ground.

Summary & Closure

Estimating the takeoff and landing distances is essential to the airplane design process. These distances are particularly critical for larger airplanes such as airliners because they typically require relatively long runways, and margins may be small. The ability of the airplane to climb out after the takeoff point and reach a safe altitude for obstacle or terrain clearance is also important. By design, the takeoff and landing airspeeds must be kept reasonably low to allow for safe flight operations, including aborted takeoffs and one-engine inoperative (OEI) conditions. The takeoff and landing distances can be significantly reduced by using high-lift devices such as flaps and slats. Good wheel brakes and reverse thrust (if available) are also crucial for minimizing landing distances.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  •  Plot some graphs to show the influence of the “ground effect” on the drag coefficient of a wing. Hint: Use reasonable values of the lift coefficient at takeoff or landing, i.e., the wing is operating near the point of stall.
  • The passenger and/or cargo load for an airliner may be limited when operating out of Denver, CO, on a hot day. Why?
  • A pilot of an airliner finds that the leading edge slats cannot be deployed during a landing. What would be the pilot’s concern regarding landing speeds and runway landing distances?
  • An emergency on a commercial airliner develops shortly after takeoff requiring the pilot to return to the airport immediately. Discuss the concerns here.

Other Useful Online Resources

To understand more about takeoff and landing performance, then follow up with some of these online resources: