52 Takeoff & Landing Performance

Introduction

The ability of airplanes to operate safely and efficiently at airports requires that they be suitably designed to achieve acceptably low takeoff and landing distances with good safety margins. The principal interest is to have sufficient runway length to the liftoff point (at a given takeoff weight and density altitude), as well as to clear any obstacles (e.g., buildings or terrain) from the initiation of the takeoff roll. It must also be ensured that the takeoff airspeeds are low enough that the pilot can abort the takeoff in an emergency and still have sufficient runway length to decelerate safely to a complete stop. As shown in Figure 1, the takeoff process appears effortless when one observes a modern airliner climbing away from the runway. However, substantial amounts of thrust and power are required to perform this aeronautical feat. Also, the airplane must be flown at a precise airspeed to maximize its angle of climb or rate of climb.

The best climb angle will be used to clear any ground obstacles, followed by building airspeed to achieve the best rate of climb to the initial cruise altitude.

The takeoff distance and airspeed at which the airplane can safely fly can be significantly reduced by using flaps and other “high-lift” devices, such as slats. The flap and slat setting selected for takeoff must balance a shorter takeoff distance against the additional drag imposed by the deployed high-lift devices. A setting that minimizes the ground roll does not necessarily maximize the initial rate of climb. For multi-engine airplanes, adequate single-engine-out (one-engine-inoperative, or OEI) performance is also essential. The loss of an engine at a critical point in the takeoff run constitutes an emergency, requiring the airplane to either stop within the remaining runway length or 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.

Learning Objectives

  • Understand the factors that affect the takeoff performance of an aircraft.
  • Appreciate the factors that can affect an airplane’s landing and stopping distances.
  • Use appropriate equations to estimate the takeoff and landing distances of an airplane.

Takeoff Performance

During a takeoff maneuver, the airplane accelerates and experiences continuously increasing airspeed. Finally, the airplane will reach sufficient airspeed to fly after covering a certain length of runway, known as the liftoff, or “unstick,” point. It is essential to keep the airplane on a straight course down the runway, which is achieved using nose-wheel steering and rudder control as airspeed increases. Then, as illustrated in Figure 2, the pilot pulls back on the elevator control to increase the pitch attitude, thereby allowing the airplane to generate sufficient lift to transition to flight and climb away from the ground. In the early stages of flight, the airplane is held at a moderate climb rate and accelerated until the best climb rate is reached. The takeoff maneuver is relatively routine, but pilots must always be prepared for unexpected emergencies, such as engine failure or other mechanical malfunctions.

The total takeoff distance is measured in terms of a ground roll, a short transition distance, and the additional airborne distance required to reach the certification screen height.

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 brings the airplane to the liftoff airspeed.
  2. The transition distance is the distance over which the airplane rotates, lifts off, and establishes the initial climb.
  3. The airborne distance required to reach the specified screen height above the takeoff surface.

The prescribed screen height depends on the applicable certification standard. Under 14 CFR Part 23, takeoff performance includes the distance required to reach 50 ft (15 m) above the takeoff surface. For transport-category airplanes under 14 CFR Part 25, the dry-runway takeoff distance is measured to a height of 35 ft above the takeoff surface. These requirements provide standardized bases for determining and reporting takeoff performance – see CFR §23.2115 and CFR §25.105 and subsequent sections.

Takeoff Analysis

Estimates of an airplane’s ground roll to liftoff and total takeoff distance can be determined using basic principles of dynamics and aerodynamics. However, solving the resulting equations typically requires numerical integration, primarily because the exact value of propulsive thrust is not known a priori; this issue also depends on whether the aircraft is a jet or a propeller aircraft. Additionally, other factors, such as the rolling friction of the wheels on the runway, may not be precisely known; however, estimates can be made for different runway surfaces, including both dry and wet conditions. Nevertheless, reasonable estimates of liftoff and takeoff distances are possible under certain assumptions. These estimates can help engineers evaluate the factors that influence the required runway lengths for a new airplane.

Consider an airplane of mass M (=W/g) acted upon by a thrust T from the propulsion system, as shown in Figure 3. There will also be a rolling resistance force, which depends on the coefficient of friction {\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 initially negligible because the dynamic pressure is very small (assuming no winds), but they increase rapidly as airspeed builds. Rolling frictional forces progressively decrease as lift develops on the wings and become zero at liftoff. After liftoff, and a positive rate of climb is established, the landing gear is retracted to reduce drag.

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, a rolling-resistance coefficient of approximately {\mu_r = 0.02} is commonly used for preliminary calculations. The value depends on the tires, airspeed, runway surface, and any contamination. Wet or contaminated surfaces can alter the rolling resistance and may also reduce the braking friction available during a rejected takeoff, so appropriate performance data must be used for the actual runway condition.

The standard aerodynamic formula gives the lift on the wing as

(2)   \begin{equation*} L = \frac{1}{2} \varrho_{\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} \varrho_{\infty} V_{\infty}^2 S C_D = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 S \left( C_{D_{0}} + \phi \frac{{C_L}^2}{\pi \, AR \, e} \right) \end{equation*}

In this case, the factor \phi accounts for the runway’s proximity to the airplane’s induced drag, i.e., the so-called “ground effect,” whose physics is explained in more detail later. While various equations have been used to represent this ground effect or “\phi factor,” one standard approximation is to use

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

where {h} is the height of the wing above the runway, and b is the wing span. 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. The airspeed increases because the thrust exceeds the sum of the friction and drag forces during the takeoff roll. 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{\big(T - \mu_r ( W - L) - D \big) g}{W} \end{equation*}

noting again that {W = M g}, so M = W/g. Therefore, after a time t from the start of the takeoff roll, the airspeed will be

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

The resultant force F depends on engine thrust, rolling friction, and drag, which depend on airspeed, {V_{\infty}}.

Equation 7 is the first integration required to estimate the flying airspeed and 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_{\textnormal{\tiny LO}} = \int_0^t V_{\infty} \, dt \end{equation*}

which can be used to determine the takeoff run and liftoff distance when 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.

Approximate Solutions

There are no closed-form solutions to the preceding equations (Eq. 7 and 8) for the liftoff distance unless making particular assumptions and approximations. Examples include constant or diminishing thrust from the propulsion system. To this end, the equation of motion as given by Eq. 6 can also be written as

(9)   \begin{equation*} \frac{1}{g} \left( \frac{dV_{\infty}}{dt} \right) = \left( \frac{T}{W} - \mu_r \right) - \frac{1}{W} \left( D - \mu_r L \right) \end{equation*}

or

(10)   \begin{equation*} \frac{1}{g} \left( \frac{dV_{\infty}}{dt} \right) = \left( \frac{T}{W} - \mu_r \right) - \frac{\displaystyle{ \frac{1}{2} \varrho_{\infty} V_{\infty}^2 S }}{W} \bigg( C_D - \mu_r \, C_L \bigg) \end{equation*}

Assume that the thrust can be written as T = T_0 - k V_{\infty}^2, where k is a constant, i.e., the thrust decreases with increasing airspeed, such as for a turbofan. In this case, then

(11)   \begin{equation*} \frac{1}{g} \left( \frac{dV_{\infty}}{dt} \right) = \left( \frac{T_0}{W} - \mu_r \right) - \frac{ V_{\infty}^2}{W} \bigg(\frac{1}{2} \varrho_{\infty} S \left( C_D - \mu_r \, C_L \right) + k \bigg) \end{equation*}

which can be written as

(12)   \begin{equation*} \frac{dV_{\infty}}{dt} = A - B V_{\infty}^2 \end{equation*}

where

(13)   \begin{equation*} A = \left( \frac{T_0}{W} - \mu_r \right) \, g \end{equation*}

and

(14)   \begin{equation*} B = \frac{g}{W} \bigg( \frac{1}{2} \varrho_{\infty} S \left( C_D - \mu_r \, C_L \right) + k \bigg) \end{equation*}

Separating the variables and integrating gives

(15)   \begin{equation*} \int \frac{dV_{\infty}}{A - B V_{\infty}^2} = \int dt \end{equation*}

which has the solution

(16)   \begin{equation*} t = \left( A \, B \right)^{-1/2} \tanh^{-1} \left( V_{\infty} \sqrt{\frac{B}{A} }\right) \end{equation*}

where \tanh^{-1} is the inverse hyperbolic tangent. Therefore, the time taken on the ground run to the point of liftoff (LO) may be obtained by replacing {V_{\infty}} with the airspeed at liftoff, which must be estimated. For the simplified analysis used here, the liftoff airspeed is assumed to be 1.2 times the airplane’s stalling airspeed in the takeoff configuration.

The corresponding ground-run length to the liftoff point can now be estimated. In this case, the equation of motion can be written as

(17)   \begin{equation*} T - \mu_r ( W - L) - D = \left( \frac{W}{g} \right) V_{\infty} \left( \frac{ d V_{\infty} }{ds} \right) \end{equation*}

where s is the distance. This equation leads to

(18)   \begin{equation*} V_{\infty} \frac{ d V_{\infty} }{ds} = A - B V_{\infty}^2 \end{equation*}

Separating the variables and integrating gives

(19)   \begin{equation*} \int \frac{ V_{\infty} d V_{\infty} }{A - B V_{\infty}^2} = \int ds \end{equation*}

which has the solution

(20)   \begin{equation*} s = -\frac{1}{2B} \ln \left( A - B V_{\infty}^2 \right) + C \end{equation*}

At s = 0 then V_{\infty} = 0, so

(21)   \begin{equation*} C = \frac{1}{2B} \ln A \end{equation*}

Therefore, for these assumptions, the liftoff distance, s_{\textnormal{\tiny LO}}, can be expressed as

(22)   \begin{equation*} s_{\textnormal{\tiny LO}} = \frac{1}{2B} \bigg( \ln A - \ln \left(A - B V_{\infty}^2 \right) \bigg) = \frac{1}{2B} \ln \left( \frac{A}{A - B V_{\infty}^2} \right) \end{equation*}

Other Approximations

Other assumptions can be used to solve the equations of motion for takeoff. However, one established approximation for estimates of the “lift-off” (LO) distance, s_{\textnormal{\tiny LO}}, for aircraft design purposes, is given by

(23)   \begin{equation*} s_{\textnormal{\tiny LO}} = \frac{1.44 \, W^2}{g \, \varrho_{\infty} \, S \, C_{L_{\rm max}} \left[ T - \bigg( D + \mu_r (W - L) \bigg)_{\rm av} \right]} \end{equation*}

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

(24)   \begin{equation*} \bigg( D + \mu_r (W - L) \bigg)_{\rm av} = \bigg( D + \mu_r (W - L) \bigg)_{0.7 V_{\textnormal{\tiny LO}}} \end{equation*}

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

(25)   \begin{equation*} V_{\textnormal{\tiny LO}} = 1.2 \, V_{\rm stall} = 1.2 \sqrt{ \frac{2 W}{\varrho_{\infty} \, S \, C_{L_{\rm max}}}} \end{equation*}

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

Further simplifications can be made to such analyses, such as assuming that the drag and rolling friction forces are small compared to the thrust force.[1] In this case,

(26)   \begin{equation*} s_{\textnormal{\tiny LO}} = \frac{1.44 \, W^2}{g \, \varrho_{\infty} \, S \, C_{L_{\rm max}} \, T} \end{equation*}

which depends only on thrust, T, so this equation provides a simple basis for identifying the primary effects on takeoff distance.

Effects of Flaps and Slats

From the preceding, it is clear that the takeoff distances are affected by the maximum lift coefficient attainable by the wing, i.e., by the value of C_{L_{\rm max}}, as shown in Figure 4. This is why flaps and slats are used on airplanes: flaps alone are more common on low-performance aircraft, whereas both flaps and slats are used on higher-performance aircraft, such as airliners. Flaps can also increase wing area when designed to move aft and deflect downward. Both flaps and slats are mechanical devices that the pilot operates as appropriate to the flight conditions.

The maximum attainable lift coefficient depends on whether the flaps or slats are retracted or deployed. When retracted, the wing is in its “clean” configuration.

Slats, however, are typically found only on higher-performance, heavier aircraft. 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 airfoil’s top surface, thereby energizing the boundary layer and delaying 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, particularly when high-lift devices are used, is generally significant, as illustrated in Figure 5.

The reduction in takeoff distances using high-lift devices is generally significant.

Numerical Solutions

The equations of motion can also be used to numerically solve the takeoff distance problem, incorporating more realistic assumptions, such as the propulsive thrust characteristics of specific engines. Airplane manufacturers carefully perform these calculations for various takeoff weights, atmospheric conditions (i.e., different density altitudes), runway conditions, and wind conditions, and then verify the results through actual flight tests. Note that the same airplane may be available from the manufacturer with different engines; therefore, the process must be completed for each airplane type and engine combination.

Representative results are shown in Figure 6, obtained by time-integrating the equations in a stepwise manner under more realistic assumptions. These types of calculations can also be repeated for an engine failure during the takeoff roll. For a twin-engine aircraft, the available thrust after the failure is approximately reduced by half. If the takeoff is rejected, the required stopping distance on the remaining runway must be calculated using wheel braking, aerodynamic drag, and reverse thrust if available. If the takeoff is continued, the remaining-engine acceleration and one-engine-inoperative (OEI) climb performance must be evaluated.

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

Runway-Surface Resistance

The rolling-resistance coefficient, \mu_r, depends strongly on the runway surface. A hard, smooth, paved runway produces relatively little resistance because the tires and runway deform only slightly. Grass, gravel, sand, mud, or other soft surfaces produce greater resistance because the tires sink into or displace the surface material. The available accelerating force is then reduced, and the takeoff ground roll increases.

Representative values for preliminary calculations are shown in the table below, although the actual rolling resistance depends on tire pressure, wheel diameter, airplane weight, surface condition, and airspeed.

Runway surface Approximate \mu_r
Hard, dry pavement 0.02
Firm, short grass 0.04-0.06
Long or wet grass 0.08-0.12
Soft ground, mud, or loose sand 0.10 or greater

The effect of the runway surface follows directly from

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

As \mu_r increases, rolling resistance becomes a larger fraction of the available thrust. Producing lift during the ground roll reduces the wheel loading, W-L, and can substantially reduce the resistance on a soft surface. However, increasing the lift coefficient also increases induced drag, so the net accelerating force becomes

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

Therefore, the most favorable ground-run lift coefficient is a compromise between reducing wheel resistance and avoiding excessive aerodynamic drag.

Using the parabolic drag polar in ground effect, i.e.,

(29)   \begin{equation*} C_D = C_{D_0}+K_g C_L^2 \end{equation*}

where

(30)   \begin{equation*} K_g=\frac{\phi}{\pi \, AR \, e} \end{equation*}

the speed-dependent resisting force can be written as

(31)   \begin{equation*} D+\mu_r(W-L) = \mu_r W + \frac{1}{2}\varrho_{\infty}V_{\infty}^2S \left(C_{D_0}+K_g C_L^2-\mu_r C_L\right) \end{equation*}

Minimizing the coefficient-dependent term with respect to C_L gives

(32)   \begin{equation*} \frac{d}{dC_L}\left(K_g C_L^2-\mu_r C_L\right)=0 \end{equation*}

so that

(33)   \begin{equation*} C_{L,\mathrm{opt}} = \frac{\mu_r}{2K_g} = \frac{\mu_r \pi \, AR \, e}{2\phi} \end{equation*}

This result shows that the lift coefficient that minimizes the combined aerodynamic and rolling resistance increases as the runway becomes softer. In practice, the attainable ground-run lift coefficient is constrained by the landing-gear geometry, flap setting, ground clearance, and directional-control requirements. Soft-field takeoff procedures exploit this same principle by reducing wheel loading while the airplane accelerates.

Temporary Thrust Augmentation

Some airplanes use a temporary source of additional thrust during takeoff. Examples include rocket-assisted takeoff, water or methanol injection, and catapult launch. In these cases, the thrust cannot be represented adequately by a single function over the entire ground roll. Instead, the takeoff analysis must be divided into separate intervals.

Suppose that an additional thrust, T_{\rm a}, is available during the initial part of the ground roll. The net accelerating force during this first interval is

(34)   \begin{equation*} F_1 = T + T_{\rm a} - D - \mu_r (W - L) \end{equation*}

and the corresponding acceleration is

(35)   \begin{equation*} \frac{dV_{\infty}}{dt} = \frac{F_1 g}{W} \end{equation*}

If the augmentation acts for a specified time t_{\rm b}, then the airspeed V_{\rm b} at which it ends must first be determined from

(36)   \begin{equation*} t_{\rm b} = \int_0^{V_{\rm b}} \frac{W} {g \left[T + T_{\rm a} - D - \mu_r (W - L)\right]} \, dV_{\infty} \end{equation*}

The distance covered during this augmented interval is then

(37)   \begin{equation*} s_1 = \int_0^{V_{\rm b}} \frac{W V_{\infty}} {g \left[T + T_{\rm a} - D - \mu_r (W - L)\right]} \, dV_{\infty} \end{equation*}

After the augmented thrust ends, the airplane continues to accelerate under its normal thrust. The net accelerating force then becomes

(38)   \begin{equation*} F_2 = T - D - \mu_r (W - L) \end{equation*}

and the remaining distance to the liftoff airspeed, V_{\textnormal{\tiny LO}}, is

(39)   \begin{equation*} s_2 = \int_{V_{\rm b}}^{V_{\textnormal{\tiny LO}}} \frac{W V_{\infty}} {g \left[T - D - \mu_r (W - L)\right]} \, dV_{\infty} \end{equation*}

The total ground-run distance is then

(40)   \begin{equation*} s_{\textnormal{\tiny LO}} = s_1 + s_2 \end{equation*}

The same method can be extended to any number of intervals in which the thrust, drag, rolling resistance, or airplane configuration changes. The terminal airspeed of one interval becomes the initial airspeed of the next. Numerical integration is usually the most convenient approach because the thrust and aerodynamic forces may vary continuously with airspeed.

Temporary thrust augmentation is generally most effective when applied early in the takeoff roll. At low airspeed, aerodynamic drag is relatively small, so a larger fraction of the additional thrust contributes directly to acceleration. The resulting increase in airspeed also carries into the subsequent unaugmented interval, reducing the remaining distance required to reach liftoff.

Effects of Winds

The effects of wind on takeoff distance must be distinguished between airspeed and ground speed. The aerodynamic forces depend on the airplane’s speed relative to the air, whereas the distance covered along the runway depends on its speed relative to the ground. If V_g is the ground speed and V_w is the headwind component, then the airspeed during the takeoff roll is approximately

(41)   \begin{equation*} V_{\infty} = V_g + V_w \end{equation*}

for a headwind. Therefore, the lift is

(42)   \begin{equation*} L = \frac{1}{2} \varrho_{\infty} \left(V_g+V_w\right)^2 S \, C_L \end{equation*}

and the drag is evaluated using the same airspeed. With a headwind, the airplane reaches the required liftoff airspeed at a lower ground speed, reducing the required ground roll. With a tailwind, the airspeed is approximately

(43)   \begin{equation*} V_{\infty}=V_g-V_w \end{equation*}

so the airplane must accelerate to a higher ground speed before reaching the required liftoff airspeed, increasing the takeoff distance. Tailwind components are particularly critical at hot-and-high conditions where the air density, \varrho_{\infty}, may already be low enough to increase takeoff distances substantially.

Takeoff Charts

At the end of performance flight testing, the airplane’s takeoff results are generalized for standard and non-standard ISA conditions and included in the flight performance charts for pilots. Figure 7 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 in the operating manual.

Using knowledge of the airplane’s weight, prevailing pressure altitude (as measured on the altimeter), outside air temperature, and wind conditions, the pilot can estimate the airplane’s anticipated takeoff distance with this type of chart. Remember that the basis of this chart is modeling and engineering data verified by testing. It is typically found in the Pilot’s Operating Handbook (POH) or Aircraft Flight Manual (AFM) for a given airplane.

While the chart looks complicated at first glance, pilots soon learn to make reasonable estimates of their takeoff performance. Notice again the significant effect of airplane weight on the takeoff distance and the impact of increasing temperature above ISA standard conditions, i.e., the impact of increasing density altitude. As previously discussed, a headwind can significantly reduce takeoff distances. The chart also suggests that takeoff into a tailwind is never advisable, as it can substantially increase takeoff distance.

For airliners and many modern airplanes with “glass cockpits,” the Flight Management Computer (FMC) or Flight Management System (FMS) assists pilots in calculating takeoff performance, including takeoff distance, critical airspeeds, and engine power settings based on real-time conditions. However, the underlying principles remain the same as those in traditional Takeoff Distance Charts; they are automated to improve efficiency and accuracy. Some airlines also require pilots to manually verify the FMC’s outputs against the charts to ensure safety and accuracy.

Reduced Thrust Takeoffs

Reduced-thrust takeoffs, also known as derated or FLEX takeoffs, are employed by many jet airliners to extend engine life, reduce maintenance costs, decrease fuel consumption, and lower emissions while maintaining ample safety margins. Derating is a fixed reduction in the maximum thrust setting, where the Flight Management Computer (FMC)  limits engine thrust to a lower, predefined level. In contrast, a FLEX takeoff involves calculating a specific power reduction based on real-time conditions, such as runway length, aircraft weight, and ambient temperature. Deliberately using less than maximum engine thrust during takeoff reduces wear on the engine(s), resulting in longer maintenance intervals and fewer overhauls. This practice can also contribute to fuel savings because the reduced thrust requirement translates into lower fuel consumption per takeoff, yielding significant cumulative savings across an airline’s fleet.

Safety remains paramount in reduced-thrust takeoffs. This option is exercised when takeoff conditions are relatively low-risk, such as a long, dry runway, a low takeoff weight, and no obstacle clearance requirements.  Performance calculations must ensure that the aircraft can still achieve the required climb rates and comply with regulations in the event of a single-engine failure during takeoff. Although this method results in slightly longer takeoff distances and potentially lower climb rates, the trade-offs are carefully managed to maintain operational efficiency and compliance with safety standards. This balance enables airlines to manage costs effectively while minimizing their environmental impact and adhering to noise-abatement procedures, particularly in sensitive areas.

Summary: Factors Affecting Takeoff Distances

In summary, regardless of the assumptions and/or approximations, the preceding results show the following outcomes:

  1. Takeoff distance increases strongly with airplane weight. Under the simplified assumption that thrust, air density, wing area, and maximum lift coefficient remain constant, the liftoff distance varies approximately with W^2. Therefore, a 50% increase in weight would increase the estimated ground-run distance by a factor of 1.5^2 = 2.25. Actual takeoff performance may differ because thrust, drag, rolling resistance, and other quantities also vary with operating conditions.
  2. The takeoff distance increases with decreasing air density (i.e., density altitude), which explains why airplanes require significantly more runway length on hot days or at high-elevation airports.
  3. The takeoff 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. Temporary thrust augmentation can reduce the takeoff ground roll. Examples include rocket-assisted takeoff, water or methanol injection, and aircraft-carrier catapults.
  5. Winds can decrease or increase takeoff distances. If there is a headwind or headwind component, takeoff distances will be reduced. Takeoff distances can increase considerably with a tailwind or its component.
  6. If reduced-thrust takeoffs are used to minimize engine wear and fuel consumption, they will result in slightly longer takeoff distances and potentially lower climb rates.

Check Your Understanding #1 – Estimating takeoff distance

A small jet aircraft has a ramp weight of 6,600 lb, and its engines have a maximum thrust of 1,200 lb. The drag polar of the aircraft in the takeoff configuration is C_D = 0.03 + 0.055 \, C_L^ {\,2}. The wing area is 160 ft2. If the winds are calm, what will be the estimated takeoff airspeed and takeoff distance? Assume that the wing stall lift coefficient, C_{L_{\rm max}}, is 1.6 and that the wing operates at C_L = 0.4 during the takeoff roll. Also, assume the runway is dry, and the aircraft is at MSL under ISA conditions.

Show solution/hide solution.

The stall airspeed of the aircraft in the takeoff configuration can be estimated using

    \[ V_{\rm stall} = \sqrt{ \frac{2 W}{\varrho_{\infty} \, S \, C_{L_{\rm max}}}} \]

For this aircraft, then

    \[ V_{\rm stall} = \sqrt{ \frac{2 \times 6{,}600}{0.002378 \times 160.0 \times 1.6} } = 140.4~\text{ft/s} \approx 83.2~\text{kts} \]

The lift-off airspeed will be 1.2 times the stall airspeed, so

    \[ V_{\textnormal{\tiny LO}} = 1.2 \, V_{\rm stall} = 1.2 \times 140.4 = 168.5~\text{ft/s} \approx 100~\text{kts} \]

The average net resistive force, R_{\rm av}, from the drag and rolling friction can be calculated using

    \[ R_{\rm av} = \bigg( D + \mu_r (W - L) \bigg)_{0.7 V_{\textnormal{\tiny LO}}} \]

where 0.7 V_{\textnormal{\tiny LO}} means that this previous quantity is evaluated at 70% of the liftoff airspeed, V_{\textnormal{\tiny LO}}. The lift on the aircraft at 0.7 V_{\textnormal{\tiny LO}} will be

    \[ L = \frac{1}{2} \varrho_{\infty} \left( 0.7 V_{\textnormal{\tiny LO}} \right)^2 S \, C_L \]

and inserting the numerical values gives

    \[ L = 0.5 \times 0.002378 \times \left( 0.7 \times 168.5 \right)^2 \times 160.0 \times 0.4 = 1{,}058.7~\text{lb} \]

The corresponding drag coefficient at C_L = 0.4 during the takeoff roll will be

    \[ C_D = 0.03 + 0.055 \, C_L^ {\,2} = 0.03 + 0.055 \times 0.4^2 = 0.0388 \]

and so the drag on the aircraft at 0.7 V_{\textnormal{\tiny LO}} will be

    \[ D = \frac{1}{2} \varrho_{\infty} \left( 0.7 V_{\textnormal{\tiny LO}} \right)^2 S \, C_D \]

Inserting the numerical values gives

    \[ D = 0.5 \times 0.002378 \times \left( 0.7 \times 168.5 \right)^2 \times 160.0 \times 0.0388 = 102.7~\text{lb} \]

The runway is dry, so {\mu_r = 0.02}. Therefore, the average resistive force is

    \[ R_{\rm av} = \bigg( D + \mu_r (W - L) \bigg)_{0.7 V_{\textnormal{\tiny LO}}} = 102.7 + 0.02 \left( 6{,}600 - 1{,}058.7 \right) = 213.5~\text{lb} \]

The takeoff distance is

    \[ s_{\textnormal{\tiny LO}} = \frac{1.44 \, W^2}{g \, \varrho_{\infty} \, S \, C_{L_{\rm max}} ( T - R_{\rm av} )} \]

and substituting all of the final values gives

    \[ s_{\textnormal{\tiny LO}} = \frac{1.44 \times 6,600^2}{32.17 \times 0.002378 \times 160.0 \times 1.6 \times \left( 1,200 - 213.5 \right)} = 3,247~\text{ft} \]

Landing Performance

As they say, what goes up must come down, so eventually the airplane needs to land. Setting up the approach to the runway requires the pilot to reduce power and airspeed to achieve a controlled rate of descent. The flaps and slats (if any) will be deployed, and the landing gear (if retractable) will be extended, as shown in Figure 8. The rate of descent is controlled by adjusting engine power and pitch attitude using the elevators. The approach continues until the aircraft is close to the ground, where the pilot flares by pulling back on the controls (up elevator) and bringing the airplane into a slightly nose-high attitude before the wheels contact the runway at the lowest possible rate of descent.

An airliner configured for landing with flaps and slats deployed and the gear down.

The final approach is flown at a prescribed airspeed safely above the stall speed, typically about 1.3\,V_{\rm stall} for the simplified analysis used here. The airplane then decelerates during the flare before touchdown, helping to minimize the landing distance. In this situation, using wing flaps and slats (if the aircraft has them) is advantageous, as it can increase the wing’s maximum lift coefficient by approximately a factor of 2. After landing, the pilot uses the rudder to maintain a straight course along the runway, then uses nose-wheel steering as airspeed decays. Reverse thrust, if available, can reduce the landing distance, especially on wet or contaminated runways, although the magnitude depends on the aircraft, runway condition, and braking effectiveness.

Analysis

Figure 9 illustrates the forces acting on an airplane during the landing rollout; they are the same as those during takeoff, except that the thrust is now zero or negative (i.e., reverse thrust is employed). The wing’s drag coefficient increases dramatically when the flaps are fully deflected and spoilers are deployed. 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 be applied only for part of the landing distance, at the pilot’s discretion and within the available runway length. Spoilers on the wings are used to reduce lift and shift weight onto the wheels, thereby enhancing rolling friction and braking during the landing roll.

The forces acting on an airplane during the landing roll.

When the airplane touches down, the pilot brings the engine thrust to idle. 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 airspeeds of many military airplanes, some may deploy a drogue parachute (or drag chute) during the landing run. Aircraft carriers quickly stop airplanes by engaging an arrestor cable across the flight deck. However, this imposes large structural loads on the airplane, resulting in uncomfortably high decelerations for the pilot and crew.

The same equations as for the takeoff apply here, but the thrust during the landing roll is now either a small positive idle-thrust value or a negative value if reverse thrust is used. Therefore,

(44)   \begin{equation*} a = \frac{dV_{\infty}}{dt} = \frac{\left[T-\mu_r(W-L)-D\right]g}{W} \end{equation*}

where T>0 for idle forward thrust and T<0 for reverse thrust. The net force must be negative for the airplane to decelerate. If the lift on the wings can be dumped by using wing spoilers, then it is possible to assume L = 0, or at least a value much smaller than W. In that case, most of the airplane’s weight is carried by the wheels, and the deceleration may be written as

(45)   \begin{equation*} \frac{dV_{\infty}}{dt} = \frac{\left(T-\mu_r W-D\right)g}{W} \end{equation*}

If idle thrust is neglected, so that T=0, then

(46)   \begin{equation*} \frac{dV_{\infty}}{dt} = -\frac{\left(\mu_r W+D\right)g}{W} = -\frac{F_R g}{W} \end{equation*}

where F_R=\mu_r W+D is the total retarding force. If reverse thrust is used, then T<0 and its magnitude must be included as an additional retarding force. If V_T is the touchdown airspeed, then the airspeed during the landing roll is obtained from

(47)   \begin{equation*} V_{\infty}(t) = V_T - \int_0^t \left(\frac{F_R}{W}\right)g\,dt \end{equation*}

and the landing-roll distance is

(48)   \begin{equation*} s_{\textnormal{\tiny L}} = \int_0^{t_{\rm stop}} V_{\infty}(t)\,dt \end{equation*}

Approximations

As with the takeoff problem, simplified solutions for landing distance are possible with certain assumptions and approximations. In this case, {\mu_r} will be greater because of the application of wheel brakes. One result often used for aircraft design purposes is that

(49)   \begin{equation*} s_{\textnormal{\tiny L}} = \frac{1.69 \, W^2}{g \varrho_{\infty} \, S \, C_{L_{\rm max}} \bigg( D + \mu_r (W - L) \bigg)_{0.7 V_{\textnormal{\tiny A}} }} \end{equation*}

where, for this simplified estimate, the touchdown airspeed is assumed to be

(50)   \begin{equation*} V_{\textnormal{\tiny T}} = 1.3 \, V_{\rm stall} = 1.3 \sqrt{ \frac{2 W}{\varrho_{\infty} \, S \, C_{L_{\rm max}}}} \end{equation*}

The factor 1.3 provides a representative margin above the stall airspeed. In an actual landing, the approach is flown at a prescribed airspeed, and the airplane normally decelerates during the flare, so the touchdown airspeed may be lower than this assumed value. In the landing configuration, flaps or other high-lift devices will increase the value of C_{L_{\rm max}} over the attainable value in the takeoff configuration.

Effects of Winds

The effects of winds on landing distances can be estimated by recognizing that a headwind reduces the ground speed corresponding to a given approach airspeed, whereas a tailwind increases it. With a headwind, the aircraft’s speed relative to the runway decreases, reducing the landing distance. As with takeoff distances, tailwind components will markedly increase landing distances because the aircraft’s effective speed relative to the runway will be higher. Again, such effects are particularly critical at “hot and high” conditions where the air density, \varrho_{\infty}, may be substantially less than at sea level.

Landing Charts

As with takeoff performance, the preceding integrations would be performed numerically (with certain assumptions), and the estimated landing distances would be validated through flight testing at different aircraft weights and atmospheric conditions, and perhaps also under varying runway conditions (e.g., dry versus wet). Along with the takeoff distance, the results will eventually be included in the airplane’s flight performance charts, which pilots can use; a simple example is shown in Figure 10. Again, this chart type will be used primarily for flight planning, such as determining whether the destination airfield has sufficient runway length.

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

Landing distance increases with airplane weight because the touchdown speed increases with wing loading and because more kinetic energy must be dissipated during the landing roll. However, it should not generally be interpreted as a simple square-of-weight dependence, because braking force, wheel loading, spoiler effectiveness, and aerodynamic drag also change with weight and speed. Distances will also increase at lower air densities, i.e., at higher altitudes, which means higher-elevation airports and hotter conditions relative to ISA. Again, the importance of high-lift devices becomes apparent: they increase the wing’s C_{L_{\rm max}} in the landing configuration, thereby reducing both the landing airspeed and the required runway length.

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 conduct numerous flight tests to verify these numerical predictions before including the final takeoff and landing performance results in the Pilot’s Operating Handbook (POH) or Aircraft Flight Manual (AFM).

Reverse Thrust

Many transport airplanes and turboprop aircraft are equipped to produce reverse thrust, but this capability is not inherent to every turbojet, turbofan, or turboprop installation. Jet-powered airplanes require a thrust-reverser system that redirects part of the exhaust or fan flow, whereas turboprop airplanes require propellers capable of moving into a negative blade-pitch range. Many aircraft, particularly military jets and smaller airplanes, do not have reverse-thrust capability.

Using reverse thrust during the landing roll can increase deceleration, reduce wheel-brake energy and wear, and provide an additional stopping force, especially at higher rollout speeds. Its effectiveness generally decreases as airspeed decreases. Reverse thrust can also increase fuel consumption, noise, maintenance requirements, and the risk of ingesting runway debris. Therefore, its availability and permitted use depend on the aircraft design, operating procedures, runway condition, and applicable noise restrictions.

Summary: Factors Affecting Landing Distances

In summary, regardless of the assumptions and/or approximations, the preceding results show the following outcomes:

  1. The landing distance increases with airplane weight because the touchdown speed and kinetic energy increase. This increase is partially offset by the greater wheel-braking force available at higher weight, so the dependence on weight is not generally quadratic. The landing distance also increases with decreasing air density (i.e., increasing density altitude), which is why airplanes require longer runways on hot days or at high-altitude airports.
  2. The landing distance is reduced by using more wing area and/or increasing C_{L_{\rm max}}. Full flaps and other high-lift devices will be deployed to minimize the landing distance.
  3. If there is a headwind or headwind component, landing distances will be reduced. If there is a tailwind or tailwind component, landing distances can increase considerably to the point that the available runway lengths may be insufficient.
  4. If available, reverse thrust can reduce landing distance, particularly on wet or contaminated runways. The reduction depends on the aircraft, reverse-thrust capability, airspeed, braking effectiveness, and runway condition.

Physics of “Ground Effect”

As previously mentioned, ground effect plays a crucial role in determining an airplane’s takeoff and landing distances. The physics behind this phenomenon requires further explanation. As an airplane nears the ground, the three-dimensional flow pattern around its wings changes, primarily affecting the formation and positions of the wing-tip vortices. The consequence of this behavior is to alter the downwash field over the wing and its induced drag, as illustrated in Figure 11.

The “ground effect” in aerodynamics manifests primarily as a reduction in downwash over the wing and in induced drag when the wing is within approximately 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 to have a higher aerodynamic aspect ratio. Ground effect is often particularly noticeable to a pilot during landing, wherein, during the landing flare, the airplane may seem 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 the ground effect, these effects are usually minor compared to the impact on the wing. The results in Figure 12, obtained with wings in wind tunnels, suggest that for less than one wing span above the ground, the “ground effect” is important and will affect the prediction of landing distances.

Line graph representing the measurements of the reduction in drag on a wing operating in ground effect.
Measurements of the reduction in drag on a wing operating in ground effect allow 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 by the equation

(51)   \begin{equation*} D_i = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 S \left( \phi \frac{{C_L}^2}{\pi \, AR e} \right) \end{equation*}

where \phi is given by the formula

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

(53)   \begin{equation*} AR_{\rm eff} = \frac{AR}{\phi} \end{equation*}

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

Check Your Understanding #2 – Estimating landing distance

The small jet aircraft considered in Example #1 has a landing weight of 5,200 lb. What will be the estimated touchdown airspeed and landing distance? Assume that the wing stall lift coefficient, C_{L_{\rm max}}, is 2.1 in the landing configuration. After touchdown, assume that the spoilers are deployed so that the residual lift is negligible and that the airplane has a representative rollout drag coefficient of C_D = 0.05. Also, assume that ground-effect aerodynamics are negligible, the winds are calm, the runway is dry, and the pilot applies full braking. The aircraft is operating at MSL under ISA conditions.

Show solution/hide solution.

The stall airspeed of the aircraft in the landing configuration can be estimated using

    \[ V_{\rm stall} = \sqrt{ \frac{2 W}{\varrho_{\infty} \, S \, C_{L_{\rm max}}}} \]

For this aircraft, then

    \[ V_{\rm stall} = \sqrt{ \frac{2 \times 5,200 }{0.002378 \times 160.0 \times 2.1} } = 114.1~\mbox{ft/s} \approx 67.6~\mbox{kts} \]

For this simplified estimate, the touchdown airspeed is assumed to be

    \[ V_{\textnormal{\tiny T}} = 1.3 \, V_{\rm stall} = 1.3 \times 114.1 = 148.3~\mbox{ft/s} \approx 87.9~\mbox{kts} \]

To estimate the landing distance, the average net resistive force, R_{\rm av}, from the aerodynamic drag and wheel braking can be calculated using

    \[ R_{\rm av} = \bigg( D + \mu_r (W - L) \bigg)_{0.7 V_{\textnormal{\tiny T}}} \]

The residual lift, L, is assumed to be negligible during the landing roll. Using the prescribed rollout drag coefficient, the drag at 0.7 V_{\textnormal{\tiny T}} will be

    \[ D = \frac{1}{2} \varrho_{\infty} \left( 0.7 V_{\textnormal{\tiny T}} \right)^2 S \, C_D \]

and inserting the numerical values gives

    \[ D = 0.5 \times 0.002378 \times \left( 0.7 \times 148.3 \right)^2 \times 160.0 \times 0.05 = 102.5~\mbox{lb} \]

The runway is dry, and full braking is applied, so {\mu_r = 0.4}. Therefore, the average resistive force during landing is

    \[ R_{\rm av} = D + \mu_r (W - 0) = 102.5 + 0.4 \times 5,200 = 2,182.5~\mbox{lb} \]

The landing distance is given by

    \[ s_{\textnormal{\tiny L}} = \frac{1.69 \, W^2}{g \, \varrho_{\infty} \, S \, C_{L_{\rm max}} R_{\rm av} } \]

and substituting the numerical values gives

    \[ s_{\textnormal{\tiny L}} = \frac{1.69 \times 5,200^2}{32.17 \times 0.002378 \times 160.0 \times 2.1 \times 2,182.5} = 814.6~\mbox{ft} \]

Summary & Closure

Estimating takeoff and landing distances is essential to aircraft design. These distances are particularly critical for larger airplanes, such as jet airliners, which typically require relatively long runways. The margins for error in these distances can be small, making precise calculations crucial. Beyond the actual distances, the airplane’s ability to climb out after the takeoff point and reach a safe altitude for obstacle or terrain clearance is also paramount. By design, takeoff and landing airspeeds must be kept reasonably low to ensure safe flight operations, including aborted takeoffs and one-engine-inoperative (OEI) conditions.

High-lift devices, such as flaps and slats, are employed to achieve the necessary low airspeeds and reduce takeoff and landing distances. These devices increase the lift generated by the wings at lower speeds, thereby enabling the airplane to take off and land from shorter runways. Additionally, effective wheel brakes and, when available, reverse thrust are crucial for minimizing landing distances. These braking mechanisms help ensure that the aircraft decelerates safely and efficiently during landing, thereby enhancing the safety and operational flexibility of larger aircraft.

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 stall point.
  • An airliner’s passenger and/or cargo load may be limited on a hot day when operating out of Denver, CO. Why?
  • If an airline pilot finds that the leading-edge slats cannot be deployed during a landing, what would be the pilot’s concern regarding landing airspeeds and runway landing distances?
  • An emergency on a commercial airliner develops shortly after takeoff, necessitating an immediate return to the airport. Discuss any concerns that the flight crew might have to address.
  • What role do environmental conditions (e.g., temperature, humidity, wind) play in aircraft takeoff and landing performance? How might these factors affect aircraft performance and the pilot’s decision-making process?
  • Discuss the significance of runway slope and its impact on aircraft takeoff and landing performance. Might an uphill or downhill slope affect the required runway length and the aircraft’s ability to take off or land safely?
  • Describe the role of flaps and other high-lift devices in reducing the stall speed during takeoff and landing. How do flaps affect takeoff and landing distances?
  • What factors might lead a pilot to reject takeoff or abort a landing attempt? What safety considerations and procedures are involved in these situations?
  • How do runway conditions, such as surface type, contamination (e.g., ice, snow, water), and tire friction, impact aircraft takeoff and landing performance? What measures are taken by airports to maintain safe runway conditions?

Other Useful Online Resources

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

  1. This assumption may be reasonable for high-performance airplanes, but it is not generally valid for all airplanes. ↵

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n40

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