translate this in persian , and i want you add equations and evaulation of formula and root of them"1. Foundational Mathematical Concepts
A. Newton's Second Law of Motion
At the heart of takeoff performance analysis lies Newton's Second Law of Motion, which articulates the relationship between the forces acting on an object, its mass, and its acceleration:

[
\sum \mathbf{F} = m \mathbf{a}
]

Where:

( \sum \mathbf{F} ) = Sum of all forces acting on the aircraft.
( m ) = Mass of the aircraft.
( \mathbf{a} ) = Acceleration of the aircraft.
B. Force Components During Takeoff
During takeoff, the primary forces include:

Thrust (( T )): Propulsion force generated by the engines.
Drag (( D )): Aerodynamic resistance opposing motion.
Lift (( L )): Vertical force enabling the aircraft to climb.
Weight (( W )): Gravitational force acting downward.
Friction (( F )): Ground resistance opposing motion, often modeled as ( F = \mu_g W ) where ( \mu_g ) is the ground friction coefficient.
C. Acceleration as a Function of Velocity
The acceleration (( a )) of the aircraft during takeoff varies with velocity due to changing aerodynamic forces:

[
a = \frac{T - D - F}{m} = \frac{T - \mu_g W - D}{W/g} = g \left( \frac{T}{W} - \mu_g - \frac{D}{W} \right )
]

Since drag (( D )) is velocity-dependent:

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

Substituting ( D ) into the acceleration equation:

[
a = g \left( \frac{T}{W} - \mu_g - \frac{1}{2} \rho V^2 \frac{S C_D}{W} \right )
]

2. Key Formulas and Their Roots
A. Takeoff Field Length (TOFL)
The Takeoff Field Length (TOFL) is the total runway distance required for an aircraft to accelerate to the Takeoff Decision Speed (( V_2 )) and achieve lift-off.

[
\text{TOFL} = \int_{V_0}^{V_2} \frac{V}{a} , dV
]

With Headwind (( V_w )) Adjustment:

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

Comprehensive Formulation Incorporating Aerodynamic Factors:

[
\text{TOFL} = \int_{0}^{V_2} \left( \frac{W}{g} \right ) \cdot \frac{V , dV}{(T - \mu_g W) - \left( C_D - \mu_g C_L \right ) \frac{1}{2} \rho S V^2 }
]

B. Lift-Off Speed (( V_{LOF} ))
Determines the minimum speed at which the aircraft can lift off the runway.

[
L = \frac{1}{2} \rho V_{LOF}^2 S C_L = W
]

Solving for ( V_{LOF} ):

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

C. Balanced Field Length (BFL)
Ensures that the runway length is sufficient for both continuing the takeoff after an engine failure and safely aborting and stopping.

[
\text{BFL} = \text{Accelerate-Go Distance} = \text{Accelerate-Stop Distance}
]

D. Takeoff Decision Speed (( V_1 ))
The speed by which the pilot must decide to abort or continue takeoff.

[
\text{S}{\text{accelerate-go}}(V_1) = \text{S}{\text{accelerate-stop}}(V_1)
]

3. Roots of Equations and Solving Methods
A. Understanding Roots in Takeoff Equations
In takeoff performance analysis, roots refer to specific velocity values where certain performance conditions are met, such as:

Lift equals weight: Determines ( V_{LOF} ).
Balanced field condition: Determines ( V_1 ).
B. Root-Finding Techniques
Due to the non-linear nature of the equations, numerical methods are essential for finding these roots. The primary techniques include:

Newton-Raphson Method:
Procedure:
[
V_{n+1} = V_n - \frac{f(V_n)}{f'(V_n)}
]
Advantages: Fast convergence near the root.
Disadvantages: Requires derivative calculation; sensitive to initial guess.
Bisection Method:
Procedure:
Choose two points ( a ) and ( b ) where ( f(a) ) and ( f(b) ) have opposite signs.
Compute midpoint ( c = \frac{a + b}{2} ).
Determine the sub-interval where the sign change occurs and repeat.
Advantages: Guaranteed convergence if the initial interval brackets a root.
Disadvantages: Slower than Newton-Raphson; requires bracketed roots.
Secant Method:
Procedure:
[
V_{n+1} = V_n - f(V_n) \cdot \frac{V_n - V_{n-1}}{f(V_n) - f(V_{n-1})}
]
Advantages: Does not require derivative; generally faster than bisection.
Disadvantages: No guaranteed convergence; can fail if ( f(V_n) = f(V_{n-1}) ).
Fixed-Point Iteration:
Procedure: Reformulate the equation as ( V = g(V) ), then iterate:
[
V_{n+1} = g(V_n)
]
Advantages: Simple to implement.
Disadvantages: Requires ( g(V) ) ensures convergence; often slower.
C. Example: Finding ( V_{LOF} ) Analytically
Given:
[
\frac{1}{2} \rho V_{LOF}^2 S C_L = W
]

Solving for ( V_{LOF} ):
[
V_{LOF} = \sqrt{ \frac{2W}{\rho S C_L} }
]

This is an analytical root, easily computed without iterative methods.

D. Example: Finding ( V_1 ) Numerically
Given the balanced field length condition:
[
S_{\text{accelerate-go}}(V_1) = S_{\text{accelerate-stop}}(V_1)
]

Define:
[
f(V_1) = S_{\text{accelerate-go}}(V_1) - S_{\text{accelerate-stop}}(V_1) = 0
]

Steps Using Newton-Raphson:

Initial Guess (( V_1^{(0)} )): Start with ( V_{LOF} \times 0.8 ).
Iterative Updates:
[
V_{1}^{(n+1)} = V_{1}^{(n)} - \frac{f(V_{1}^{(n)})}{f'(V_{1}^{(n)})}
]
Convergence: Repeat until ( |f(V_1^{(n+1)})| ) is below a set tolerance (e.g., ( 10^{-3} )).
Considerations:

Derivative (( f'(V_1) )): Often estimated numerically as:
[
f'(V_1) \approx \frac{f(V_1 + \Delta V) - f(V_1 - \Delta V)}{2 \Delta V}
]
Software Tools: Utilizing software like MATLAB, Python (with NumPy/SciPy), or Excel can automate iterations and enhance accuracy.
4. Comparative Analysis of the Three Sources
A. Exact (Precise) Methods
Aspect	Ajo Kumar	Trevor M. Young	Jan Roskam
TOFL Formula	(\text{TOFL} = \int_{0}^{V_2} \frac{V}{a} , dV)	(\text{TOFL} = \int_{V_w}^{V_{LOF}} \frac{V - V_w}{a_g} , dV)	Similar to Young with more detailed segmentation
Acceleration Expression (a)	( a = \frac{g (T - \mu_g W - D)}{W} )	( a_g = g \left( \frac{T}{W} - \mu_g - \frac{C_{D_g} q}{W/S} - \phi \right ) )	Similar to Young, includes balance and wind corrections
Numerical Integration Technique	Simpson's rule with speed partitioning	Explicit use of Simpson's rule for S_G and S_TR	Detailed Simpson's rule with correction factors for wind
Assumptions	Constant acceleration within partitions	Small angles, constant coefficients within speed intervals	Small angles, accurate averaging at ( \frac{V_{LOF}}{\sqrt{2}} )
Additional Calculations	Sum over speed partitions for ( S_G )	Includes transition distance ( S_{TR} )	Segmented into ( S_{NGR} + S_R ), incorporating wind and rotation phases
Insights:

All sources utilize integral calculus for precise TOFL calculation, acknowledging the non-linear relationship between acceleration and velocity.
Kumar and Young share similar foundational formulas, with Young explicitly dividing the ground run and transition distances.
Roskam builds upon Young’s methods by adding further segmentation, considering wind and varying conditions more comprehensively.
B. Statistical Methods
Aspect	Ajo Kumar	Trevor M. Young	Jan Roskam
Statistical Method Usage	Minimal/Implicated	Extensive using FAR 25 TOP correlations	Similar to Young, tailored for specific regulatory standards
Primary Formula	N/A	( S_{\text{TOFL}} = 37.5 \times \text{TOP}_{25} )	Similar, with adjustments for All Engines Operating (AEO) conditions
Parameters Emphasized	N/A	Thrust-to-weight, wing loading, ( C_{L_{max}} ), density ratio	Similar parameters, emphasizes segmented ground run
Applicability	Not focused on statistical methods	Primarily for preliminary design and regulatory compliance	Similar to Young, useful for rapid estimations aligned with regulations
Insights:

Young and Roskam provide clear empirical correlations (TOP parameters) aligned with FAR 25 regulations, facilitating quick TOFL estimations without intricate force balance integrations.
Kumar does not emphasize statistical methods, focusing more on exact analytical and numerical integrations.
C. Approximate and Analytical Methods
Aspect	Ajo Kumar	Trevor M. Young	Jan Roskam
Use of Average Acceleration	Yes: ( S_G \approx \frac{V_{LOF}^2}{2 \bar{a}} )	Yes: Mean acceleration at ( \frac{V_R}{\sqrt{2}} )	Yes: Similar to Young, detailed for different phases
Simplified TOFL/Formulas	( S_G \approx \frac{V_{LOF}^2}{2 \bar{a}} )	( S_R \approx \frac{(V_{LOF} \pm V_w)^2}{2g(\ldots)} )	Detailed balanced field length formulas incorporating ( S_{NGR} + S_R )
Energy Balance Approach	Brief discussion	Explicit use in transition and climb distances	Comprehensive application, including climb and transition phases
Trade-off	Faster computation with acceptable accuracy	Balance between simplicity and inclusion of various phases	Similar to Young, with added detail for phases and environmental factors
Assumptions	Steady-state acceleration; constant coefficients within intervals	Small angles; constant acceleration at mean speed	Small angles; accurate averaging; comprehensive consideration of phases
Insights:

All sources employ average acceleration assumptions to simplify TOFL calculations, balancing computational feasibility with reasonable accuracy.
Roskam extends approximate methods by integrating all takeoff phases, ensuring a holistic approximation of TOFL.
D. Balanced Field Length Calculations
Aspect	Ajo Kumar	Trevor M. Young	Jan Roskam
Balanced Field Length Formula	Implicit through comprehensive exact methods	Explicit by combining ( S_R + S_{RL} )	Explicit segmentation into ( S_{NGR} + S_R ) with wind considerations
Inclusion of Wind and Ground Slope	General integration includes wind effects	Incorporates wind in rotation and ground run segments	Detailed consideration of wind and runway slope in ( S_{NGR} )
Empirical Data Usage	Limited to exact methods	Uses empirical timing and speed data for rotation ( t_{\text{rotate}} )	Similar to Young, with detailed segmentation for balanced field compliance
Regulatory Compliance	Not explicitly covered	Ensures compliance through FAR 25 TOP correlations	Detailed alignment with balanced field requirements, enhancing regulatory compliance
Insights:

Roskam provides the most detailed and explicit treatment of balanced field length, ensuring that all necessary runways segments account for emergency scenarios.
Young incorporates balanced field length indirectly by combining specific runway segments and using empirical data.
Kumar does not explicitly focus on balanced field length but lays the groundwork through comprehensive force integration.
5. Additional Formulas and Enhancements from Broader Knowledge
To deepen the comparative analysis, let's introduce additional relevant formulas and concepts that are pivotal in takeoff performance analysis:

A. Rate of Climb Equation
Once airborne, the aircraft must climb to a certain height, typically to clear obstacles.

[
\text{Rate of Climb} = \frac{W - L - D + T}{m}
]

Where:

( L ) = Lift
( D ) = Drag
( T ) = Thrust
( W ) = Weight
( m ) = Mass
Approximation for Climb Angle (( \theta )):

[
\theta = \arcsin\left( \frac{R}{V} \right ) \approx \frac{R}{V}
]

Where:

( R ) = Vertical speed
( V ) = Forward speed
B. Power Required for Climb
[
P_{\text{climb}} = (T - D) V
]

Where:

( V ) = Airspeed
( (T - D) ) = Net thrust available for climb
C. Flaps and Slats Impact on Lift and Drag
Deploying flaps and slats increases ( C_L ) while also increasing ( C_D ):

[
C_{L_{\text{flaps}}} = C_{L_{\text{clean}}} + \Delta C_L
]
[
C_{D_{\text{flaps}}} = C_{D_{\text{clean}}} + \Delta C_D
]

D. Ground Roll Distance (( S_{\text{GR}} ))
[
S_{\text{GR}} = \frac{V_R^2}{2 a_{\text{GR}}}
]

Where:

( V_R ) = Rotation speed
( a_{\text{GR}} ) = Ground acceleration
E. Transition and Climb Distances
[
S_{\text{TR}} = R_{\text{TR}} \sin{\theta_{\text{CL}}}
]
[
S_{\text{CL}} = \frac{h_{\text{screen}} - h_{\text{TR}}}{\tan{\theta_{\text{CL}}}}
]

Where:

( R_{\text{TR}} ) = Transition radius
( \theta_{\text{CL}} ) = Climb angle
( h_{\text{screen}} ) = Clearance height (e.g., 35 ft, 50 ft)
( h_{\text{TR}} ) = Height at transition phase
F. Energy-Based Takeoff Distance
[
\int (T - D) , ds = \frac{1}{2} m (V_{\text{LOF}}^2 - V_0^2) + m g h
]

Where:

( ds ) = Differential distance increment
This approach focuses on the energy changes required for takeoff, providing an alternative to force-based integrations.

6. Comparative Summary and Insights
A. Similarities Across Sources
Foundational Principles: All sources base their analysis on Newton's Second Law, considering thrust, drag, lift, weight, and friction.
Integral Calculations: Use integral calculus to account for the varying acceleration as the aircraft speeds up.
Numerical Integration: Employ similar numerical methods (Simpson's Rule, Trapezoidal Rule) for integrating complex, non-linear functions.
Balanced Field Length: Younger and Roskam provide explicit methods for calculating balanced field length, implicit or foundational in Kumar.
Environmental Factors: All consider wind speed and runway slope, though the depth and method of incorporation vary.
B. Differences Across Sources
Methodological Complexity:
Roskam offers the most detailed segmentation, considering multiple phases and environmental factors comprehensively.
Young strikes a balance between precision and applicability, integrating empirical methods with exact calculations.
Kumar emphasizes exact numerical integrations, with limited focus on statistical methods.
Use of Statistical Methods:
Young and Roskam employ empirical correlations aligned with FAR 25 (TOP parameters) for quick estimates.
Kumar does not prioritize statistical methods, focusing more on exact integrations for precise TOFL calculations.
Balanced Field Length Treatment:
Roskam provides a comprehensive approach, segmenting into non-ground run and rotation distances with wind considerations.
Young incorporates balanced field length through combined ground run segments and empirical correlations.
Kumar lays the foundation through integrative exact methods but doesn't explicitly define balanced field length.
Analytical vs. Numerical Focus:
Roskam and Young provide both analytical insights and numerical methods, facilitating a versatile application.
Kumar is more numerically precise, relying on computational methods for exact solutions.
C. Practical Implications
Design Phases:
Empirical Methods (Young and Roskam): Ideal for preliminary design phases where quick estimates are needed.
Exact Numerical Integrations (Kumar): Suited for detailed performance analyses during advanced design stages.
Regulatory Compliance:
Roskam and Young ensure compliance with FAR standards through their balanced field length and empirical correlation approaches.
Kumar provides the numerical precision necessary for validating regulatory compliance but lacks inherent empirical adjustments.
Computational Requirements:
Roskam and Young offer formulas that can be manually computed with moderate computational tools.
Kumar's methods generally require more advanced computational tools due to their reliance on iterative numerical methods.
7. Enhanced Comparative Example with All Formulas
Let's undertake a detailed comparative example using the provided formulas from each source. We'll calculate the Takeoff Field Length (TOFL) for a hypothetical aircraft under standard conditions.

Given Parameters:
Weight (( W )): 600,000 N
Thrust (( T )): 700,000 N
Drag Coefficient (( C_D )): 0.3
Lift Coefficient (( C_L )): 1.5
Air Density (( \rho )): 1.225 kg/m³
Wing Area (( S )): 500 m²
Ground Friction Coefficient (( \mu_g )): 0.04
Wind Speed (( V_w )): 10 m/s (headwind)
LOF Speed (( V_{LOF} )): ( \sqrt{ \frac{2W}{\rho S C_L} } )
Rotation Speed (( V_R )): 80 m/s
Climb Clearance Height (( h_{\text{screen}} )): 35 ft (10.668 m)
Acceleration due to Gravity (( g )): 9.81 m/s²
A. Calculating ( V_{LOF} )
[
V_{LOF} = \sqrt{ \frac{2W}{\rho S C_L} } = \sqrt{ \frac{2 \times 600,000}{1.225 \times 500 \times 1.5} } = \sqrt{ \frac{1,200,000}{918.75} } \approx \sqrt{1307.74} \approx 36.15 , \text{m/s}
]

B. Exact Method Comparisons
1. Ajo Kumar
TOFL Formula:

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

Assuming ( V_2 = V_{LOF} ), and substituting ( a ):

[
\text{TOFL} = \int_{0}^{36.15} \frac{V}{g \left( \frac{T}{W} - \mu_g - \frac{1}{2} \rho S C_D V^2 / W \right )} , dV
]

Numerical Integration Approach:

Given the integral's complexity, use Simpson's rule with ( n = 4 ) (for simplicity):

[
\text{TOFL} \approx \frac{\Delta V}{3} [f(V_0) + 4f(V_1) + 2f(V_2) + 4f(V_3) + f(V_4)]
]

Where ( \Delta V = \frac{36.15}{4} = 9.0375 , \text{m/s} ), and ( V_0 = 0 ), ( V_1 = 9.0375 ), ( V_2 = 18.075 ), ( V_3 = 27.1125 ), ( V_4 = 36.15 ).

Function ( f(V) = \frac{V}{a(V)} ):

[
a(V) = 9.81 \left( \frac{700,000}{600,000} - 0.04 - \frac{1}{2} \times 1.225 \times 500 \times 0.3 \times V^2 / 600,000 \right )
]

Simplify:

[
a(V) = 9.81 \left( 1.1667 - 0.04 - \frac{0.18375 V^2}{600,000} \right ) = 9.81 \left( 1.1267 - 3.0625 \times 10^{-7} V^2 \right )
]

Calculate ( a(V) ) and ( f(V) ) at each ( V_i ):

At ( V_0 = 0 , \text{m/s} ):
[
a(0) = 9.81 \times 1.1267 \approx 11.048 , \text{m/s}^2
]
[
f(0) = \frac{0}{11.048} = 0
]

At ( V_1 = 9.0375 , \text{m/s} ):
[
a(9.0375) = 9.81 \times \left(1.1267 - 3.0625 \times 10^{-7} \times 81.68 \right ) \approx 9.81 \times 1.1267 \approx 11.048 , \text{m/s}^2
]
[
f(9.0375) = \frac{9.0375}{11.048} \approx 0.818 , \text{m/s}
]

At ( V_2 = 18.075 , \text{m/s} ):
[
a(18.075) = 9.81 \times \left(1.1267 - 3.0625 \times 10^{-7} \times 326.72 \right ) \approx 9.81 \times 1.1267 \approx 11.048 , \text{m/s}^2
]
[
f(18.075) = \frac{18.075}{11.048} \approx 1.634 , \text{m/s}
]

At ( V_3 = 27.1125 , \text{m/s} ):
[
a(27.1125) = 9.81 \times \left(1.1267 - 3.0625 \times 10^{-7} \times 734.27 \right ) \approx 9.81 \times 1.1267 \approx 11.048 , \text{m/s}^2
]
[
f(27.1125) = \frac{27.1125}{11.048} \approx 2.453 , \text{m/s}
]

At ( V_4 = 36.15 , \text{m/s} ):
[
a(36.15) = 9.81 \times \left(1.1267 - 3.0625 \times 10^{-7} \times 1307.74 \right ) \approx 9.81 \times 1.1267 \approx 11.048 , \text{m/s}^2
]
[
f(36.15) = \frac{36.15}{11.048} \approx 3.273 , \text{m/s}
]

Simpson's Rule Calculation:

[
\text{TOFL} \approx \frac{9.0375}{3} [0 + 4(0.818) + 2(1.634) + 4(2.453) + 3.273] = 3.0125 \times [0 + 3.272 + 3.268 + 9.812 + 3.273] = 3.0125 \times 19.625 \approx 59.1 , \text{meters}
]

Note: The accurate value would require precise calculation of ( a(V) ) at each ( V_i ), considering slight velocity-dependent changes. The above example simplifies ( a(V) ) for illustration.

2. Trevor M. Young
TOFL Formula:

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

Acceleration ( a_g ):

[
a_g = g \left( \frac{T}{W} - \mu_g - \frac{(C_{D_g} - \mu_g C_{L_g}) q}{W/S} - \phi \right )
]

Assuming negligible ( \phi ), similar to the earlier expression.

Numerical Integration Using Simpson's Rule:

Divide ( V_w = 10 , \text{m/s} ) to ( V_{LOF} = 36.15 , \text{m/s} ) into manageable intervals (e.g., ( n = 4 )).

Function ( f(V) = \frac{V - V_w}{a_g} ):

[
f(V) = \frac{V - 10}{9.81 \left( \frac{700,000}{600,000} - 0.04 - \frac{1}{2} \times 1.225 \times 500 \times 0.3 \times V^2 / 600,000 \right ) }
]

Repeat the Simpson's rule steps similar to Kumar's approach.

Simplified Calculation:

Given similar dynamics, TOFL should be in the same order of magnitude as Kumar's, adjusted for headwind.

Approximate TOFL:

[
\text{TOFL} \approx 37.5 \times \text{TOP}_{25}
]

Where:
[
\text{TOP}{25} = \frac{(W/S){TO}}{\sigma \cdot C_{L_{max_{TO}}} \cdot (T/W)_{TO}} = \frac{1,200}{1 \times 1.5 \times 1.1667} \approx 686.96
]
[
\text{TOFL} \approx 37.5 \times 686.96 \approx 25,761 , \text{meters}
]

Note: The TOP formula provided seems to overestimate; ensure alignment with specific source definitions.

C. Jan Roskam
TOFL Formula:

Roskam's approach is similar but includes more detailed segmentation and environmental factors:

[
\text{TOFL} = \frac{(V_{LOF} \pm V_w)^2}{2g \left( \frac{T}{W} - \mu_g - \frac{(C_D - \mu_g C_L) q}{W/S} - \phi \right ) }
]

Balanced Field Length Segmentation:

[
S_G = S_{NGR} + S_R
]

Where:

( S_{NGR} ) = Non-Ground Run Distance
( S_R ) = Rotation Distance with wind considerations
Numerical Integration with Simpson's Rule:

Roskam applies Simpson's rule similarly but adjusts for each segment's specific conditions, enhancing accuracy.

Final TOFL Calculation:

[
\text{TOFL} = \frac{(36.15 + 10)^2}{2 \times 9.81 \times \text{Adjusted Acceleration}} \approx \frac{(46.15)^2}{19.62 \times 10.048} \approx \frac{2127.22}{196.99} \approx 10.82 , \text{meters}
]

Note: Accurate calculation requires precise adjustment of acceleration based on all factors, likely resulting in a larger, more realistic TOFL."