translate to persian "1. Rasakam:
Climb Distance to the Obstacle (S_{CL})
Root Formula:
[
S_{CL} = \left(\frac{V_2 + V_{INCL}}{2}\right) \left(\frac{h_{screen} - h_{INCL}}{\left(\frac{V_2 + V_{INCL}}{2}\right) \gamma}\right)
]
Development:
The average speed during the climb phase is (\frac{V_2 + V_{INCL}}{2}).
The vertical distance to be covered is (h_{screen} - h_{INCL}).
The climb angle is (\gamma).
The horizontal distance (S_{CL}) can be found using the relationship:
[
S_{CL} = \frac{h_{screen} - h_{INCL}}{\gamma}
]
Statistical Method for Takeoff Distance Prediction (FAR 25)
Root Formula:
[
S_{TOFL} = 37.5 \cdot TOP_{25}
]
[
TOP_{25} = \frac{\left(\frac{W}{S}\right){TO}}{\sigma \cdot C{L_{max_{TO}}} \cdot \left(\frac{T}{W}\right)_{TO}}
]
Development:
(S_{TOFL}) is the total field length required for takeoff.
(TOP_{25}) is the takeoff parameter, which is a dimensionless number that combines several key parameters.
(\left(\frac{W}{S}\right)_{TO}) is the wing loading at takeoff.
(\sigma) is the air density ratio.
(C_{L_{max_{TO}}}) is the maximum lift coefficient at takeoff.
(\left(\frac{T}{W}\right)_{TO}) is the thrust-to-weight ratio at takeoff.
The formula is derived from empirical data and is used to predict takeoff distance under various conditions.
Approximate Method for Ground Distance Calculation (S_{NGR})
Root Formula:
[
a_g = a_{gV=0} - \left(a_{gV=0} - a_{gV=V_R}\right) \frac{V^2}{V_R^2}
]
[
a_{gV=0} = g \left(\frac{T_{V=0}}{W} - \mu_g - \phi\right)
]
[
a_{gV=V_R} = g \left(\left(\frac{T_{V=V_R}}{W} - \mu_g\right) - \left(\frac{C_D - \mu_g C_L}{2W/S}\right) \rho V_R^2 - \phi\right)
]
[
S_{NGR} = \frac{V_R^2}{2a_{g_{ave}}}
]
[
a_{g_{ave}} = a_{gV=0} \times \left(\frac{1 - \frac{a_{gV=V_R}}{a_{gV=0}}}{\ln\left(\frac{a_{gV=0}}{a_{gV=V_R}}\right)}\right)
]
Development:
(a_g) is the ground acceleration, which varies with velocity (V).
(a_{gV=0}) is the initial ground acceleration at (V = 0).
(a_{gV=V_R}) is the ground acceleration at the rotation speed (V_R).
(a_{g_{ave}}) is the average ground acceleration, calculated using the logarithmic mean.
The ground distance (S_{NGR}) is derived from the kinematic equation:
[
S_{NGR} = \frac{V_R^2}{2a_{g_{ave}}}
]
2. T. M. Yang:
Takeoff Distance from Brake Release to Rotation (s_R)
Root Formula:
[
s_R = \int_{V_w}^{V_R} \frac{V - V_w}{a} dV
]
Development:
The distance (s_R) is the integral of the time taken to reach the rotation speed (V_R) from the wind speed (V_w).
The acceleration (a) is a function of velocity (V).
The equation accounts for the effect of wind on the takeoff distance.
Simplified Equation Using Average Acceleration:
[
s_R = \frac{1}{2a} (V_R - V_w)^2
]

Development:
The average acceleration (a) is used to simplify the integral.
The distance (s_R) is derived from the kinematic equation:
[
s_R = \frac{1}{2a} (V_R - V_w)^2
]
Energy Method for Takeoff Distance to 35 Feet (s_{L35})
Root Formula:
[
s_{L35} = \left(\frac{W}{(F_N - D){avg}}\right) \left(\frac{V{35}^2 - V_{LOF}^2}{2g} + h_{sc}\right)
]
Development:
The work done by the net force ((F_N - D)_{avg}) is equal to the change in kinetic and potential energy.
The work done is given by:
[
U = \int (F_N - D) ds
]
The change in kinetic energy is:
[
\Delta KE = \frac{1}{2} m (V_{35}^2 - V_{LOF}^2)
]
The change in potential energy is:
[
\Delta PE = mgh_{sc}
]
Equating the work done to the change in energy:
[
(F_N - D){avg} s{L35} = \frac{1}{2} m (V_{35}^2 - V_{LOF}^2) + mgh_{sc}
]
Solving for (s_{L35}):
[
s_{L35} = \left(\frac{W}{(F_N - D){avg}}\right) \left(\frac{V{35}^2 - V_{LOF}^2}{2g} + h_{sc}\right)
]
3. Ajay Kumar:
Ground Run Distance (TOFL)
Root Formula:
[
\mathrm{TOFL} = \int_{0}^{V_2} \frac{V}{a} dV
]
[
\mathrm{TOFL} = \int_{V_w}^{V_2} \frac{V - V_w}{a} dV
]
Development:
The distance (\mathrm{TOFL}) is the integral of the time taken to reach the takeoff speed (V_2) from zero or wind speed (V_w).
The acceleration (a) is a function of velocity (V).
Detailed Equation:
[
\mathrm{TOFL} = \int_{0}^{V_2} \left(\frac{W}{g}\right) \cdot \left(\frac{V dV}{\left(T - \mu W\right) - \left(C_D - \mu C_L\right) \frac{1}{2} \rho S V^2}\right)
]

Development:
The acceleration (a) is given by Newton's second law:
[
a = \frac{\sum F}{m} = \frac{g}{W} \sum F
]
The net force (\sum F) is:
[
\sum F = T - \mu W - \left(C_D - \mu C_L\right) \frac{1}{2} \rho S V^2
]
Substituting (\sum F) into the integral:
[
\mathrm{TOFL} = \int_{0}^{V_2} \left(\frac{W}{g}\right) \cdot \left(\frac{V dV}{\left(T - \mu W\right) - \left(C_D - \mu C_L\right) \frac{1}{2} \rho S V^2}\right)
]
Numerical Integration:
[
S_G = \sum_{\Delta V_i}^{\Delta V_f} \Delta S_g
]
[
\Delta S_G = \frac{V_f^2 - V_i^2}{2\bar{a}} = \frac{(V_f - V_i) V_{ave}}{\bar{a}}
]

Development:
The ground run distance (S_G) is approximated by summing the incremental distances (\Delta S_g).
The average acceleration (\bar{a}) is used to simplify the calculation:
[
\bar{a} = \left(\frac{g}{W}\right) \left[\left(\bar{T} - \mu W\right) - \left(\bar{C_D} - \mu \bar{C_L}\right) Sq\right]
]
The incremental distance (\Delta S_g) is:
[
\Delta S_G = \frac{V_f^2 - V_i^2}{2\bar{a}} = \frac{(V_f - V_i) V_{ave}}{\bar{a}}
]
Simplified Equation Using Average Acceleration:
[
S_{G_{LO}} = \frac{V_{LO}^2}{2\bar{a}} = \frac{W V_{LO}^2}{2g \left[\left(T - \mu W\right) - \left(C_D - \mu C_L\right) S q\right]{0.707 V{LO}}}
]

Development:
The average acceleration (\bar{a}) is used to simplify the calculation:
[
\bar{a} = \left(\frac{g}{W}\right) \left[\left(T - \mu W\right) - \left(C_D - \mu C_L\right) S q\right]{0.707 V{LO}}
]
The ground run distance (S_{G_{LO}}) is:
[
S_{G_{LO}} = \frac{V_{LO}^2}{2\bar{a}} = \frac{W V_{LO}^2}{2g \left[\left(T - \mu W\right) - \left(C_D - \mu C_L\right) S q\right]{0.707 V{LO}}}
]"