translate to persian and add explanation to equations and formula in the text with solving methode "1. Ground Effect
Ajo Kumar:
Kumar integrates ground effect implicitly within the overall aerodynamic force equations. The influence of ground effect on lift and drag is considered through modified lift ((C_L)) and drag ((C_D)) coefficients in the acceleration equation:

[
a = \frac{g}{W} \left[ (T - \mu W) - (C_D - \mu C_L) S q - W \delta \right]
]

Here, (C_L) and (C_D) are adjusted to account for ground effect without explicit separate formulas.

Dr. Jan Roskam:
Roskam provides a more explicit treatment of ground effect by directly modifying the lift and drag coefficients. Ground effect enhances the effective aspect ratio, thereby increasing the lift coefficient and reducing induced drag. The formulas used are:

[
C_{L_g} = C_{L_{ag}} (\alpha - \alpha_0 - \Delta \alpha_{0_g})
]

[
C_{D_g} = C_{D_0} + \lambda K C_L^2
]

Where:

(C_{L_g}) and (C_{D_g}) are the lift and drag coefficients in ground effect.
(C_{L_{ag}}) is the adjusted lift coefficient accounting for ground effect.
(\alpha_0) is the zero-lift angle of attack, and (\Delta \alpha_{0_g}) represents the reduction in this angle due to ground effect.
(\lambda) and (K) are empirical coefficients related to ground effect drag reduction.
Trevor M. Young:
Young employs an empirical adjustment for drag under ground effect, enhancing the alpha-dependent terms to reflect decreased induced drag:

[
C_{D_g} = C_{D_0} + \lambda K C_L^2
]

Additionally, Young defines the change in zero-lift angle of attack due to ground effect:

[
\Delta \alpha_{0_g} = \left( \frac{t}{c} \right) \left[ -0.11771 \left( \frac{1}{\left( \frac{h}{c} \right)^2} \right) + \frac{3.56551}{\frac{h}{c}} \right]
]

Where:

(t/c) is the thickness-to-chord ratio.
(h/c) represents the nondimensional height of the wing above the ground.
These adjustments directly influence (C_{L_g}) and (C_{D_g}), aligning theoretical models with empirical observations.
2. Thrust and Thrust Vectoring
Ajo Kumar:
Kumar models thrust ((T)) as a constant value during takeoff but acknowledges that thrust can vary with speed and configuration. The general thrust equation is integrated into the acceleration formula:

[
a = \frac{g}{W} \left[ T - \mu W - (C_D - \mu C_L) S q - W \delta \right]
]

Dr. Jan Roskam:
Roskam maintains thrust as a primary driving force but introduces considerations for thrust vectoring. Thrust vectoring is incorporated by decomposing the thrust vector into components aligned and perpendicular to the flight path:

[
T_x = T \cos(\theta_T)
]
[
T_y = T \sin(\theta_T)
]

Where (\theta_T) is the thrust deflection angle. These components are then integrated into the force balance equations:

[
\frac{dV}{dt} = \frac{g}{W} \left[ T_x - D_g - \mu_g W + \mu_g L_g - W \phi \right]
]

Trevor M. Young:
Young extends the thrust modeling to include empirical thrust reduction with increasing speed and thrust vectoring effects. The thrust variation is modeled as:

[
\frac{F_N}{F_{N_0}} = 1 - k_1 V + k_2 V^2
]

Where:

(F_N) is the thrust at speed (V).
(F_{N_0}) is the static thrust (at (V = 0)).
(k_1) and (k_2) are empirically determined coefficients.
For thrust vectoring, Young adjusts the effective thrust used for acceleration:

[
T_{\text{effective}} = T \cos(\theta_{\text{deflection}})
]

This reduction in effective thrust is incorporated into the acceleration and takeoff distance calculations.

3. Drag Modeling
Ajo Kumar:
Drag is modeled using the standard aerodynamic drag equation, incorporating both profile drag and induced drag components:

[
D = \frac{1}{2} \rho V^2 S C_D
]

Where (C_D) is further broken down:

[
C_D = C_{D_0} + K C_L^2
]

Here, (C_{D_0}) is the zero-lift drag coefficient, and (K) is the induced drag factor.

Dr. Jan Roskam:
Roskam emphasizes the impact of ground effect on drag by modifying the drag coefficient:

[
C_{D_g} = C_{D_0} + \lambda K C_L^2
]

This modified drag coefficient accounts for reduced induced drag due to ground effect. Additionally, Roskam integrates induced drag reduction explicitly:

[
\Delta C_{D_{\text{ig}}} = -\sigma' \frac{C_L^2}{\pi A}
]

Where (\sigma') is the ground effect drag coefficient adjustment factor, and (A) is the aspect ratio.

Trevor M. Young:
Young refines the drag model under ground effect by incorporating empirical adjustments:

[
C_{D_g} = C_{D_0} + \lambda K C_L^2
]

He also accounts for changes in drag during rotation and lift-off phases by adjusting (C_D) based on speed and configuration, ensuring the drag accurately reflects different takeoff stages.

4. Lift Modeling
Ajo Kumar:
Lift during ground roll is modeled using the standard lift equation:

[
L = \frac{1}{2} \rho V^2 S C_L
]

Lift is incorporated into the acceleration equation by adjusting the rolling friction:

[
a = \frac{g}{W} \left[ T - \mu (W - L) - (C_D - \mu C_L) S q - W \delta \right]
]

Dr. Jan Roskam:
Roskam incorporates ground effect into the lift coefficient:

[
C_{L_g} = C_L \frac{C_{L_{ag}}}{C_{L_u}} \Delta \alpha_{0_g}
]

Where (C_{L_u}) is the lift coefficient outside ground effect. This adjusted lift coefficient is used to calculate lift:

[
L_g = \frac{1}{2} \rho V^2 S C_{L_g}
]

Trevor M. Young:
Young further refines lift modeling by considering the effective aspect ratio in ground effect and its impact on lift:

[
C_{L_g} = C_{L_{ag}} (\alpha - \alpha_0 - \Delta \alpha_{0_g})
]

He also addresses lift dynamics post-lift-off during the climb phase, ensuring lift calculations remain accurate as the aircraft transitions from ground roll to flight.

5. Wind Components and Their Influence
Ajo Kumar:
Kumar adjusts the takeoff field length (TOFL) calculation to account for wind components:

[
\mathrm{TOFL} = \int_{0}^{V_2} \frac{V - V_w}{a} , dV
]

Where (V_w) is the wind component along the runway (positive for headwind, negative for tailwind).

Dr. Jan Roskam:
Roskam incorporates wind directly into the ground speed equation:

[
V_G = V - V_w
]

He then adjusts the acceleration and TOFL calculations accordingly:

[
\frac{dV}{dt} = \frac{g}{W} \left[ T - D_g - \mu_g W + \mu_g L_g - W \phi \right]
]

[
\mathrm{TOFL} = \int_{V_w}^{V_{LOF}} \frac{V - V_w}{a_g} , dV
]

Trevor M. Young:
Young models wind influence by adjusting both the speed and drag coefficients:

[
S_R = \frac{(V_{LOF} \pm V_w)^2}{2g \left[ \left( \frac{T}{W} - \mu \right) - \left( \frac{C_D - \mu C_L}{Sq} \right) - \phi \right]}
]

He also accounts for wind shear corrections:

[
V_w = V_{w50} \times F_{\text{shear}}
]

Where (V_{w50}) is the wind speed measured at 50 feet, and (F_{\text{shear}}) is an empirical correction factor.

6. Rolling Friction and Runway Slope
Ajo Kumar:
Rolling friction is modeled as:

[
F_{\text{fr}} = \mu (W - L)
]

Where ( \mu ) is the rolling friction coefficient and (L) is the lift force. The runway slope ((\delta)) introduces an additional component to the acceleration equation:

[
a = \frac{g}{W} \left[ T - \mu (W - L) - (C_D - \mu C_L) S q - W \delta \right]
]

Dr. Jan Roskam:
Roskam separates the force balance to explicitly account for runway slope effects:

[
T - D_g - \mu_g W + \mu_g L_g - W \phi = \frac{W}{g} \frac{dV}{dt}
]

Here, (\phi) represents the runway slope expressed as a ratio (e.g., 1% slope as 0.01). The slope affects both the drag and the net acceleration of the aircraft.

Trevor M. Young:
Young incorporates runway slope into the frictional forces:

[
\sum F = F_N - D - \mu (W \cos{\gamma_G} - L) - W \sin{\gamma_G}
]

Where (\gamma_G) is the runway slope angle. Approximating for small angles ((\sin{\gamma_G} \approx \gamma_G), (\cos{\gamma_G} \approx 1)) simplifies the equation:

[
\sum F = F_N - D - \mu (W - L) - W \gamma_G
]

This formulation directly ties runway slope to the net accelerating force, influencing both acceleration and takeoff distance.

7. Takeoff Phases: Ground Roll, Rotation, and Lift-off
Ajo Kumar:
Kumar views the takeoff as a continuous phase where acceleration varies with velocity. The rotation and lift-off are treated implicitly within the numerical integration of velocity increments.

Dr. Jan Roskam:
Roskam distinctly separates the takeoff into phases—ground roll, rotation, and lift-off. Each phase has tailored equations reflecting the changing dynamics:

Ground Roll: [
\frac{dV}{dt} = \frac{g}{W} \left[ T - D_g - \mu_g W + \mu_g L_g - W \phi \right]
]
Rotation: Introduces angular dynamics and modifies lift and drag as the nose gear lifts.
Lift-off: Transition to climbing, adjusting lift to overcome weight and incorporating climb angle dynamics.
Trevor M. Young:
Young further refines these phases by incorporating rejected takeoff (RTO) scenarios and braking dynamics. He provides specific equations for each phase:

Ground Roll to Rotation: [
S_R = \frac{(V_{LOF} \pm V_w)^2}{2g \left[ \left( \frac{T}{W} - \mu \right) - \left( \frac{C_D - \mu C_L}{Sq} \right) - \phi \right]}
]
Rotation and Lift-off: Adjusts lift and drag coefficients dynamically as the aircraft transitions to flight, integrating climb dynamics: [
T \cos(\alpha + \Phi_T) - D_g - W \sin{\gamma} = W g \frac{dV}{dt}
] [
T \sin(\alpha + \Phi_T) + L_g - W \cos{\gamma} = C.F. \cdot W g V \gamma
]
Lift-off Speed Determination: [
V_{LOF} = \sqrt{\frac{2W}{\rho S C_{L_g}}}
]
8. Rejected Takeoff (RTO) Scenarios
Ajo Kumar:
Kumar's primary focus is on the continuous takeoff phase without explicit consideration of RTO scenarios. However, the numerical integration approach can be adapted for RTO by altering force balances post-engine failure.

Dr. Jan Roskam:
Roskam addresses RTO by integrating brake performance and engine failure into the takeoff distance calculations. He outlines scenarios where an engine failure alters the thrust and drag dynamics, requiring modified acceleration and distance computations.

Trevor M. Young:
Young provides the most comprehensive treatment of RTO, detailing explicit formulas for calculating post-engine failure distances:

[
s_{RTO} = \frac{V_1^2}{2 a_{0,1}} + \frac{V_{35}^2 - V_1^2}{2 a_{1,35}} \left( \frac{N}{N-1} \right)
]

Where:

( V_1 ) = Decision speed at which a takeoff should be aborted
( V_{35} ) = Speed at 35 feet altitude
( a_{0,1} ) = Acceleration from rest to ( V_1 )
( a_{1,35} ) = Acceleration from ( V_1 ) to ( V_{35} )
( N ) = Number of engines
This formulation accounts for the reduced thrust from failing engines and adapts the acceleration accordingly.

9. Brake Performance and Stopping Distance
Ajo Kumar:
While Kumar acknowledges the role of braking in takeoff (especially in RTO scenarios), his primary focus remains on acceleration and distance to liftoff. Detailed brake performance equations are not extensively covered.

Dr. Jan Roskam:
Roskam incorporates braking performance indirectly through rolling friction and adjustable thrust in RTO analyses. Specific brake force models are integrated into the force balance equations during RTO.

Trevor M. Young:
Young offers a detailed examination of brake performance, introducing formulas that incorporate brake efficiency and maximum brake coefficients:

[
\mu_B = \eta_B \mu_{\text{max}}
]

[
s_B = \frac{V^2}{2 \mu_B g}
]

Where:

( \mu_B ) = Brake coefficient
( \eta_B ) = Brake efficiency factor (e.g., 0.3 for on-off systems, 0.8 for fully modulated systems)
( \mu_{\text{max}} ) = Maximum brake coefficient
He also discusses systems to prevent wheel lock-up and optimize brake performance under various conditions, integrating these factors into RTO distance calculations.

10. Wind Shear and Wind Correction Factors
Ajo Kumar:
Kumar adjusts wind speed in TOFL calculations by incorporating ( V_w ) as a constant bias in velocity integration. However, specific wind shear corrections are not explicitly detailed.

Dr. Jan Roskam:
Roskam acknowledges wind shear but primarily accounts for it through empirical correction factors applied to measured wind speeds:

[
V_w = V_{w50} \times F_{\text{shear}}
]

Where:

( V_{w50} ) = Wind speed measured at 50 feet
( F_{\text{shear}} ) = Empirical wind shear correction factor
Trevor M. Young:
Young provides a more nuanced treatment of wind shear, integrating it into the wind component calculations and its impact on ground speed (( V_G )):

[
V_G = V - V_w
]

He further adjusts the TOFL and acceleration equations to account for varying wind conditions across different altitudes, enhancing the accuracy of performance predictions under dynamic wind profiles.

11. Extended Takeoff Distance Calculations
Ajo Kumar:
Kumar's extended calculations involve numerical summation over velocity increments, allowing for the inclusion of all dynamic factors without the need for closed-form solutions.

[
S_G = \sum_{\Delta V_i}^{\Delta V_f} \frac{W (V_f - V_i) V_{\text{avg}}}{g \left[ (\overline{T} - \mu W) - (\overline{C_D} - \mu \overline{C_L}) S q \right]}
]

Dr. Jan Roskam:
Roskam extends takeoff distance calculations by breaking them into ground roll, rotation, and lift-off phases, applying specific formulas and integration techniques to each phase for higher precision.

[
\mathrm{TOFL} = S_{\text{Ground Roll}} + S_{\text{Rotation}} + S_{\text{Lift-off}}
]

Trevor M. Young:
Young provides comprehensive formulas that encompass all takeoff phases, including RTO scenarios, wind corrections, and brake performance:

[
\mathrm{TOFL} = \int_{0}^{V_{LOF}} \frac{W V}{g \left[ T - \frac{1}{2} \rho V^2 S (C_{D_g} - \mu C_L) - \mu (W - L) - W \delta \right]} dV + s_{RTO} + s_B
]

This integrated approach ensures that all contributing factors are accounted for in the final takeoff distance estimation.

12. Additional Formulas and Enhancements
To provide a deeper mathematical insight, several additional formulas from the excerpts are integrated:

Lift-off Speed in Ground Effect:

[
V_{LOF} = \sqrt{\frac{2W}{\rho S C_{L_g}}}
]

Climb Phase Dynamics:

After lift-off, the aircraft enters the climb phase, where lift must overcome weight and induce climb:

[
T \cos(\alpha + \Phi_T) - D_g - W \sin{\gamma} = W g \frac{dV}{dt}
]

[
T \sin(\alpha + \Phi_T) + L_g - W \cos{\gamma} = C.F. \cdot W g V \gamma
]

Where:

(\Phi_T) = Thrust vectoring angle
(\gamma) = Climb angle rate
(C.F.) = Centripetal force coefficient
Wind Shear Correction:

[
V_w = V_{w50} \times F_{\text{shear}}
]

Takeoff Roll Acceleration Including Ground Effect and Wind:

[
a = \frac{g}{W} \left[ T - \frac{1}{2} \rho V^2 S (C_{D_g} - \mu C_L) - \mu (W - L) - W \delta \right]
]

Brake Performance Integration in RTO:

[
s_{RTO} = \frac{V_1^2}{2 a_{0,1}} + \frac{V_{35}^2 - V_1^2}{2 a_{1,35}} \left( \frac{N}{N-1} \right)
]

Where:

(N) = Number of engines
Total Takeoff Distance Incorporating All Factors:

[
\mathrm{TOFL} = S_G + S_{RTO} + S_B
]

Where:

(S_G) = Ground roll distance
(S_{RTO}) = Rejected takeoff distance
(S_B) = Braking distance
Topical Comparison of Formula Development
The following sections illustrate how each author develops and applies formulas for specific takeoff performance topics, highlighting their unique approaches and methodologies.

A. Ground Effect
Ajo Kumar:

Approach: Implicit integration within overall acceleration equations.
Key Formula: [
a = \frac{g}{W} \left[ (T - \mu W) - (C_D - \mu C_L) S q - W \delta \right]
]
Development: Adjusts lift and drag coefficients within a unified force balance, avoiding separate ground effect equations.
Dr. Jan Roskam:

Approach: Explicit modification of aerodynamic coefficients to account for ground effect.
Key Formulas: [
C_{L_g} = C_{L_{ag}} (\alpha - \alpha_0 - \Delta \alpha_{0_g})
] [
C_{D_g} = C_{D_0} + \lambda K C_L^2
]
Development: Ground effect is directly incorporated into lift and drag coefficients, enabling precise adjustments based on aerodynamic models.
Trevor M. Young:

Approach: Empirical adjustments to drag and lift coefficients under ground effect.
Key Formulas: [
C_{D_g} = C_{D_0} + \lambda K C_L^2
] [
\Delta \alpha_{0_g} = \left( \frac{t}{c} \right) \left[ -0.11771 \left( \frac{1}{\left( \frac{h}{c} \right)^2} \right) + \frac{3.56551}{\frac{h}{c}} \right]
]
Development: Utilizes empirical coefficients to adjust lift and drag, ensuring ground effect is accurately reflected based on wing geometry and proximity to the ground.
B. Thrust and Engine Performance
Ajo Kumar:

Approach: Models thrust as a constant force during takeoff.
Key Formula: [
a = \frac{g}{W} \left[ T - \mu W - (C_D - \mu C_L) S q - W \delta \right]
]
Development: Incorporates thrust directly into the acceleration equation, assuming steady thrust output.
Dr. Jan Roskam:

Approach: Incorporates thrust vectoring by decomposing thrust into directional components.
Key Formulas: [
T_x = T \cos(\theta_T)
] [
T_y = T \sin(\theta_T)
] [
\frac{dV}{dt} = \frac{g}{W} \left[ T_x - D_g - \mu_g W + \mu_g L_g - W \phi \right]
]
Development: Adjusts thrust direction for vectoring, allowing simulation of different thrust angles during takeoff.
Trevor M. Young:

Approach: Models thrust variation with speed and incorporates thrust vectoring effects.
Key Formulas: [
\frac{F_N}{F_{N_0}} = 1 - k_1 V + k_2 V^2
] [
T_{\text{effective}} = T \cos(\theta_{\text{deflection}})
]
Development: Uses empirical relationships to account for thrust reduction as speed increases, integrating these adjustments into acceleration and distance calculations.
C. Drag Modeling
Ajo Kumar:

Approach: Utilizes standard aerodynamic drag equations with lift-induced drag consideration.
Key Formula: [
C_D = C_{D_0} + K C_L^2
] [
D = \frac{1}{2} \rho V^2 S C_D
]
Development: Separates zero-lift drag and induced drag, integrating them into the overall drag force.
Dr. Jan Roskam:

Approach: Explicitly modifies drag coefficients to reflect ground effect.
Key Formulas: [
C_{D_g} = C_{D_0} + \lambda K C_L^2
] [
\Delta C_{D_{\text{ig}}} = -\sigma' \frac{C_L^2}{\pi A}
]
Development: Adjusts drag coefficients based on ground effect, accounting for reduced induced drag through empirical factors.
Trevor M. Young:

Approach: Enhances drag modeling by incorporating ground effect and empirical adjustments.
Key Formulas: [
C_{D_g} = C_{D_0} + \lambda K C_L^2
] [
D_g = \frac{1}{2} \rho V^2 S C_{D_g}
]
Development: Applies empirical coefficients to refine drag predictions under ground effect, ensuring alignment with observed performance.
D. Lift Modeling
Ajo Kumar:

Approach: Uses standard lift equations, integrating lift into the overall acceleration formula.
Key Formula: [
L = \frac{1}{2} \rho V^2 S C_L
] [
a = \frac{g}{W} \left[ T - \mu (W - L) - (C_D - \mu C_L) S q - W \delta \right]
]
Development: Incorporates lift to adjust rolling friction, effectively reducing the net resisting force as lift increases.
Dr. Jan Roskam:

Approach: Adjusts lift coefficients to account for ground effect and phase-specific dynamics.
Key Formula: [
C_{L_g} = C_{L_{ag}} (\alpha - \alpha_0 - \Delta \alpha_{0_g})
] [
L_g = \frac{1}{2} \rho V^2 S C_{L_g}
]
Development: Ground effect modifications ensure lift calculations accurately reflect the near-ground aerodynamic environment.
Trevor M. Young:

Approach: Factors in ground effect and phase transitions to dynamically adjust lift.
Key Formula: [
C_{L_g} = C_L \frac{C_{L_{ag}}}{C_{L_u}} \Delta \alpha_{0_g}
] [
L_g = \frac{1}{2} \rho V^2 S C_{L_g}
]
Development: Enhances lift modeling by incorporating empirical adjustments and geometric factors, ensuring precise lift predictions during takeoff phases.
E. Takeoff Field Length (TOFL) Calculation
Ajo Kumar:

Approach: Employs numerical summation over velocity increments to estimate TOFL.
Key Formula: [
\mathrm{TOFL} = \sum \frac{W (V_f - V_i) \overline{V}}{g \left[ (\overline{T} - \mu W) - (\overline{C_D} - \mu \overline{C_L}) S q \right]}
] [
S_G \approx \frac{V_{LO}^2}{2 \bar{a}}
]
Development: Utilizes discretized velocity steps to integrate acceleration and accumulate distance, balancing computational efficiency with accuracy.
Dr. Jan Roskam:

Approach: Applies numerical integration techniques like Simpson’s Rule to compute TOFL.
Key Formula: [
\mathrm{TOFL} = \int_{0}^{V_2} \frac{V}{a} , dV
]
Development: Utilizes Simpson’s Rule for higher precision in integrating velocity-dependent acceleration, especially under varying aerodynamic conditions.
Trevor M. Young:

Approach: Derives analytical expressions for TOFL, incorporating empirical adjustments and multiple takeoff phases.
Key Formula: [
\mathrm{TOFL} = \int_{0}^{V_{LOF}} \frac{W V}{g \left[ T - \frac{1}{2} \rho V^2 S (C_{D_g} - \mu C_L) - \mu (W - L) - W \delta \right]} , dV + s_{RTO} + s_B
]
Development: Combines analytical integration for continuous acceleration phases with additional distances from RTO and braking, providing a comprehensive TOFL model.
F. Rejected Takeoff (RTO) Scenarios
Ajo Kumar:

Approach: Treats RTO implicitly by altering force balances during numerical integration.
Key Formula: While specific RTO equations are not provided, the numerical approach allows for the modification of thrust and drag parameters post-engine failure within the summation: [
S_G = \sum \frac{W (V_f - V_i) \overline{V}}{g \left[ (\overline{T} - \mu W) - (\overline{C_D} - \mu \overline{C_L}) S q \right]}
]
Development: Adapts the overall force equation to reflect engine failure by reducing (T) and adjusting (C_D) accordingly during the summation process.
Dr. Jan Roskam:

Approach: Incorporates RTO by adjusting thrust and braking forces based on engine failure points.
Key Formula: [
\mathrm{TOFL_{RTO}} = \int_{V_{EF}}^{V_1} \frac{V}{a_{RTO}} , dV + \text{Braking Distance}
] Where (V_{EF}) is the engine failure speed and (V_1) is the decision speed.
Development: Separates RTO into distinct stages, calculating distances before and after engine failure, and integrating brake performance into the total distance.
Trevor M. Young:

Approach: Provides comprehensive formulas for RTO, accounting for engine failure at decision speed ( V_1 ).
Key Formulas: [
s_{RTO} = s_{0,1} + s_{1,35}
] Where: [
s_{0,1} = \frac{V_1^2}{2 a_{0,1}}
] [
s_{1,35} = \frac{V_{35}^2 - V_1^2}{2 a_{1,35}} \left( \frac{N}{N-1} \right)
]
Development: Explicitly calculates the rejected takeoff distance by partitioning it into acceleration before and after engine failure, incorporating the reduced thrust from failing engines.
G. Brake Performance and Stopping Distance
Ajo Kumar:

Approach: Acknowledges braking as a factor in RTO scenarios but does not provide detailed brake performance equations.
Key Consideration: Incorporates rolling friction adjustments in the acceleration equations which indirectly affect braking.
Dr. Jan Roskam:

Approach: Integrates braking forces within RTO calculations by modifying the overall force balance during aborted takeoff.
Key Formula: [
\sum F = T - D_g - \mu_g (W - L_g) - W \phi - F_{\text{brakes}}
]
Development: Accounts for brake forces as additional resistive forces in the RTO distance computations.
Trevor M. Young:

Approach: Provides detailed equations for braking performance, incorporating brake efficiency and maximum coefficients.
Key Formulas: [
\mu_B = \eta_B \mu_{\text{max}}
] [
s_B = \frac{V^2}{2 \mu_B g}
]
Development: Models braking distance by calculating the deceleration induced by brake forces, factoring in system efficiency and maximum achievable braking coefficients.
H. Wind Shear and Corrections
Ajo Kumar:

Approach: Adjusts TOFL calculations by integrating ( V_w ) as a constant wind component but does not explicitly model wind shear.
Key Formula: [
\mathrm{TOFL} = \int_{0}^{V_2} \frac{V - V_w}{a} , dV
]
Development: Treats wind shear as part of the overall wind component affecting ground speed.
Dr. Jan Roskam:

Approach: Applies empirical wind shear correction factors to measured wind speeds.
Key Formula: [
V_w = V_{w50} \times F_{\text{shear}}
]
Development: Adjusts wind speed based on shear factors to more accurately reflect wind conditions along the takeoff path.
Trevor M. Young:

Approach: Provides a nuanced treatment of wind shear by integrating it into ground speed calculations and adjusting acceleration accordingly.
Key Formula: [
V_G = V - V_w
] [
\mathrm{TOFL} = \int_{0}^{V_{LOF}} \frac{V - V_w}{a} , dV
]
Development: Incorporates wind shear through dynamic adjustments of ( V_w ) across different altitude segments during takeoff.
I. Additional Considerations and Enhanced Formulas
Ajo Kumar:

Numerical Summation Integration: [
S_G = \sum \frac{W (V_f - V_i) \overline{V}}{g \left[ (\overline{T} - \mu W) - (\overline{C_D} - \mu \overline{C_L}) S q \right]}
] This method facilitates the accumulation of takeoff distance by incrementing velocity in discrete steps.
Dr. Jan Roskam:

Phased Integration for TOFL: [
\mathrm{TOFL} = S_{\text{Ground Roll}} + S_{\text{Rotation}} + S_{\text{Lift-off}}
] Each segment is calculated using specific acceleration and force balance equations tailored to the phase.
Trevor M. Young:

Comprehensive TOFL with RTO and Braking: [
\mathrm{TOFL} = \int_{0}^{V_{LOF}} \frac{W V}{g \left[ T - \frac{1}{2} \rho V^2 S (C_{D_g} - \mu C_L) - \mu (W - L) - W \delta \right]} , dV + s_{RTO} + s_B
] This integrated formula accounts for continuous acceleration, rejected takeoff distances, and braking performance, providing a holistic takeoff distance calculation.
Summary of Formula Development by Topic
Below is a topical comparison outlining which formulas each book uses for specific takeoff performance factors:

1. Ground Effect
Kumar: Incorporates ground effect implicitly within lift and drag coefficients. [
a = \frac{g}{W} \left[ (T - \mu W) - (C_D - \mu C_L) S q - W \delta \right]
]
Roskam: Explicitly modifies lift and drag coefficients. [
C_{L_g} = C_{L_{ag}} (\alpha - \alpha_0 - \Delta \alpha_{0_g})
] [
C_{D_g} = C_{D_0} + \lambda K C_L^2
]
Young: Employs empirical adjustments for lift and drag under ground effect. [
C_{D_g} = C_{D_0} + \lambda K C_L^2
] [
\Delta \alpha_{0_g} = \left( \frac{t}{c} \right) \left[ -0.11771 \left( \frac{1}{\left( \frac{h}{c} \right)^2} \right) + \frac{3.56551}{\frac{h}{c}} \right]
]
2. Thrust and Engine Performance
Kumar: Models thrust as a constant input in the acceleration equation. [
a = \frac{g}{W} \left[ (T - \mu W) - (C_D - \mu C_L) S q - W \delta \right]
]
Roskam: Decomposes thrust for vectoring. [
T_x = T \cos(\theta_T)
] [
T_y = T \sin(\theta_T)
] [
\frac{dV}{dt} = \frac{g}{W} \left[ T_x - D_g - \mu_g W + \mu_g L_g - W \phi \right]
]
Young: Models thrust variation with speed and incorporates thrust vectoring effects. [
\frac{F_N}{F_{N_0}} = 1 - k_1 V + k_2 V^2
] [
T_{\text{effective}} = T \cos(\theta_{\text{deflection}})
]
3. Drag Modeling
Kumar: Standard aerodynamic drag with lift-induced drag. [
C_D = C_{D_0} + K C_L^2
] [
D = \frac{1}{2} \rho V^2 S C_D
]
Roskam: Adjusted drag coefficients for ground effect. [
C_{D_g} = C_{D_0} + \lambda K C_L^2
] [
\Delta C_{D_{\text{ig}}} = -\sigma' \frac{C_L^2}{\pi A}
]
Young: Empirical drag adjustments under ground effect. [
C_{D_g} = C_{D_0} + \lambda K C_L^2
] [
D_g = \frac{1}{2} \rho V^2 S C_{D_g}
]
4. Lift Modeling
Kumar: Standard lift equation integrated into acceleration. [
L = \frac{1}{2} \rho V^2 S C_L
]
Roskam: Adjusted lift coefficients for ground effect. [
C_{L_g} = C_{L_{ag}} (\alpha - \alpha_0 - \Delta \alpha_{0_g})
] [
L_g = \frac{1}{2} \rho V^2 S C_{L_g}
]
Young: Empirical lift adjustments under ground effect and during climb. [
C_{L_g} = C_L \frac{C_{L_{ag}}}{C_{L_u}} \Delta \alpha_{0_g}
] [
L_g = \frac{1}{2} \rho V^2 S C_{L_g}
]
5. Wind Components and Their Influence
Kumar: Incorporates wind speed as a constant bias in TOFL integration. [
\mathrm{TOFL} = \int_{0}^{V_2} \frac{V - V_w}{a} , dV
]
Roskam: Applies wind shear corrections to measured wind speeds. [
V_w = V_{w50} \times F_{\text{shear}}
]
Young: Integrates wind shear into ground speed and TOFL calculations. [
V_G = V - V_w
] [
\mathrm{TOFL} = \int_{0}^{V_{LOF}} \frac{V - V_w}{a} , dV
]
6. Rolling Friction and Runway Slope
Kumar: Models rolling friction within the acceleration equation: [
F_{\text{fr}} = \mu (W - L)
] [
a = \frac{g}{W} \left[ (T - \mu W) - (C_D - \mu C_L) S q - W \delta \right]
]
Roskam: Explicitly includes runway slope in the force balance: [
\frac{dV}{dt} = \frac{g}{W} \left[ T - D_g - \mu_g W + \mu_g L_g - W \phi \right]
]
Young: Incorporates slope as an additional resistive force: [
\sum F = F_N - D - \mu (W - L) - W \gamma_G
] [
a = \frac{g}{W} \left[ T - \frac{1}{2} \rho V^2 S (C_{D_g} - \mu C_L) - \mu (W - L) - W \delta \right]
]"