Category
Aerospace Engineering
Aerospace engineering formulas and calculators for aerodynamics, propulsion, and flight mechanics in aircraft and spacecraft design. Includes lift, drag, thrust, and orbital trajectory equations.
98Formulas
2Subcategories
Updated Jul 2026
General
Minimum Orbital Inclination from Launch Latitude
i_min = latitude of launch site
General
Service Ceiling Criterion
R/C_service = 0.508 m/s (100 ft/min)
General
Kepler's Equation
M = E - e*sin(E)
General
Absolute Ceiling Relation (Zero Rate of Climb)
R/C = 0 at Absolute Ceiling
Propulsion
Propeller Aircraft Range (Simplified)
R = (η_p/c_p) * (C_L/C_D) * ln(W_i/W_f)
Propulsion
Propeller Aircraft Endurance (Simplified)
E = (η_p / c_p) * (C_L^1.5/C_D) * sqrt(2*ρ*S) * (W_i^-0.5 - W_f^-0.5) / (-0.5)
General
Hohmann Transfer Total Delta-V
Δv_total = Δv1 + Δv2
General
Orbit Equation (Semi-Latus Rectum Form)
r = p / (1 + e*cos(θ))
General
Specific Orbital Angular Momentum
h = r * v * cos(γ)
General
Specific Orbital Energy
ε = v^2/2 - μ/r = -μ/(2a)
General
Orbital Plane Change Delta-V
Δv = 2 * v * sin(Δi/2)
General
Stagnation Point Convective Heating Rate (Simplified)
q̇ = k * sqrt(ρ/R_n) * V^3
Propulsion
Choked (Sonic) Mass Flow Rate
mdot = (p_c * A_t / sqrt(T_c)) * sqrt(γ/R) * (2/(γ+1))^((γ+1)/(2*(γ-1)))
Propulsion
Ideal Rocket Nozzle Exit Velocity
V_e = sqrt((2*γ/(γ-1)) * R * T_c * (1 - (p_e/p_c)^((γ-1)/γ)))
Propulsion
Overall Propulsion Efficiency
η_o = η_th * η_p
Propulsion
Brayton Cycle Thermal Efficiency
η_th = 1 - 1/(r_p^((γ-1)/γ))
Propulsion
Propulsive Efficiency
η_p = 2*V_0 / (V_e + V_0)
Propulsion
Ramjet Thrust Equation (Simplified)
F = mdot * (V_e - V_0)
Propulsion
Turbojet Thrust Equation
F = mdot * (V_e - V_0) + A_e * (p_e - p_0)
Propulsion
Thrust Specific Fuel Consumption (TSFC)
TSFC = mdot_f / T
Propulsion
Fuel Flow Rate
mdot_f = TSFC * T
Propulsion
Specific Range
SR = V / (c * T)
Aerodynamics
Bank Angle for Coordinated Turn
φ = arctan(V^2 / (g * R))
Aerodynamics
Load Factor in Coordinated Turn
n = 1 / cos(φ)
General
Aircraft Roll Moment of Inertia (Simplified Wing Model)
I_xx ≈ (m_wing * b^2) / 12
General
Centrifugal Force on Rotor Blade Element
F_c = m * Ω^2 * r
General
Rotor Solidity Ratio
σ = (N_b * c) / (π * R)
General
Momentum Theory Induced Velocity (Hover)
v_i = sqrt(T / (2 * ρ * A))
General
Figure of Merit (Rotor Hover Efficiency)
FM = P_ideal / P_actual
General
Helicopter Disk Loading
DL = W / A
General
Helicopter Rotor Blade Tip Speed
V_tip = Ω * R
Propulsion
Propeller Power Coefficient
C_P = P / (ρ * n^3 * D^5)
Propulsion
Propeller Thrust Coefficient
C_T = T / (ρ * n^2 * D^4)
Propulsion
Propeller Advance Ratio
J = V / (n * D)
Propulsion
Propeller Efficiency
η_p = T * V / P_shaft
Propulsion
Power Available (Propeller-Driven Aircraft)
P_avail = η_p * P_shaft
Propulsion
Power Required for Level Flight
P_req = D * V = (C_D * q * S) * V
General
Landing Ground Roll Distance (Simplified)
s_L ≈ 1.69 * W^2 / (g * ρ * S * C_Lmax * D_avg)
General
Takeoff Ground Roll Distance (Simplified)
s_TO ≈ 1.44 * W^2 / (g * ρ * S * C_Lmax * T)
General
Fuel Weight Fraction
W_f/W_0 = 1.06 * (1 - W_x/W_0)
General
Empty Weight Fraction
W_e/W_0 = W_empty / W_0
Aerodynamics
Torsional Divergence Speed (Simplified)
V_d = sqrt(K_θ / (q_coeff * e * c * C_Lα * S))
Aerodynamics
Wing Root Bending Moment (Elliptical Load, Simplified)
M_root ≈ (4/(3π)) * L * (b/2)
Aerodynamics
Simplified Flutter Speed Estimate
V_f ≈ ω_θ * b * sqrt((I_α/(ρ*b^4)) * (something))
Aerodynamics
V-n Diagram Positive Limit Load Factor
n_max = 2.1 + 24000/(W + 10000)
Aerodynamics
Dutch Roll Natural Frequency (Approx.)
ω_d ≈ sqrt((N_β * q̄ * S * b) / I_z)
Aerodynamics
Phugoid Mode Period (Approx.)
T_ph ≈ 2π * sqrt(2) * V / g
Aerodynamics
Short Period Natural Frequency (Approx.)
ω_sp ≈ sqrt((-M_α * q̄ * S * c̄) / I_y)
Aerodynamics
Pitching Moment Coefficient
C_m = M / (q * S * c̄)
Aerodynamics
Static Margin
SM = (x_np - x_cg) / c̄
Aerodynamics
Neutral Point Location
x_np = (C_L,α,wb * x_ac,wb + C_L,α,t * η * (S_t/S) * x_ac,t) / (C_L,α,wb + C_L,α,t * η * (S_t/S))
Aerodynamics
Center of Pressure Location
x_cp = x_ac - (C_m,ac / C_L)
Aerodynamics
Kutta-Joukowski Theorem
L' = ρ * V * Γ
Aerodynamics
Blasius Laminar Boundary Layer Thickness
δ = 5.0 * x / sqrt(Re_x)
Aerodynamics
Turbulent Flat Plate Skin Friction Coefficient
C_f = 0.074 / Re_x^(1/5)
Aerodynamics
Laminar Flat Plate Skin Friction Coefficient
C_f = 1.328 / sqrt(Re_x)
Aerodynamics
Sutherland's Law (Viscosity)
μ = μ0 * (T/T0)^(3/2) * (T0+S)/(T+S)
Aerodynamics
Stagnation Pressure
p0 = p * (1 + ((γ-1)/2) * M^2)^(γ/(γ-1))
Aerodynamics
Stagnation Temperature
T0 = T * (1 + ((γ-1)/2) * M^2)
Aerodynamics
Speed of Sound (Ideal Gas)
a = sqrt(γ * R * T)
Propulsion
Area-Mach Number Relation (Isentropic Nozzle)
(A/A*)^2 = (1/M^2) * ((2/(γ+1))*(1+((γ-1)/2)*M^2))^((γ+1)/(γ-1))
Aerodynamics
Critical Mach Number Relation
C_p,cr = (2/(γ*M_cr^2)) * (((1+((γ-1)/2)*M_cr^2)/(1+(γ-1)/2))^(γ/(γ-1)) - 1)
Aerodynamics
Prandtl-Glauert Compressibility Correction
C_p = C_p0 / sqrt(1 - M^2)
Aerodynamics
Normal Shock Downstream Mach Number
M2^2 = (1 + ((γ-1)/2)*M1^2) / (γ*M1^2 - (γ-1)/2)
Aerodynamics
Normal Shock Pressure Ratio
p2/p1 = 1 + (2γ/(γ+1)) * (M1^2 - 1)
Aerodynamics
Isentropic Density Ratio
ρ0/ρ = (1 + ((γ-1)/2) * M^2)^(1/(γ-1))
Aerodynamics
Isentropic Pressure Ratio
p0/p = (1 + ((γ-1)/2) * M^2)^(γ/(γ-1))
Aerodynamics
Isentropic Temperature Ratio
T0/T = 1 + ((γ-1)/2) * M^2
Propulsion
Total Impulse
I_t = ∫ F dt ≈ F_avg * t_b
Propulsion
Characteristic Velocity
c* = p_c * A_t / mdot
Propulsion
Nozzle Thrust Coefficient
C_F = F / (p_c * A_t)
Propulsion
Effective Exhaust Velocity
v_e = Isp * g0
Propulsion
Rocket Mass Ratio
MR = m0 / mf
General
Geostationary Orbit Radius
r = (μ * T^2 / (4π^2))^(1/3)
General
Hohmann Transfer Time of Flight
t = π * sqrt(a_t^3 / μ)
General
Vis-Viva Equation
v = sqrt(μ * (2/r - 1/a))
General
Circular Orbital Velocity
v = sqrt(μ / r)
General
Hohmann Transfer First Burn Delta-V
Δv1 = sqrt(μ/r1) * (sqrt(2*r2/(r1+r2)) - 1)
Propulsion
Tsiolkovsky Rocket Equation
Δv = v_e * ln(m0 / mf)
Propulsion
Thrust-to-Weight Ratio
T/W = Thrust / Weight
Propulsion
Breguet Endurance Equation (Jet)
E = (1/c) * (L/D) * ln(W_i / W_f)
Propulsion
Breguet Range Equation (Jet)
R = (V / c) * (L/D) * ln(W_i / W_f)
Propulsion
Rate of Climb
R/C = (T - D) * V / W
Aerodynamics
Radius of Turn (Coordinated)
R = V^2 / (g * tan(φ))
Aerodynamics
Stall Speed
V_stall = sqrt(2*W / (ρ * S * C_Lmax))
Aerodynamics
Lift-to-Drag Ratio
L/D = C_L / C_D
Aerodynamics
Wing Loading
W/S = Weight / S
Aerodynamics
Oswald Efficiency Factor Relation
e = C_L^2 / (π * AR * C_Di)
Aerodynamics
Induced Drag Coefficient
C_Di = C_L^2 / (π * e * AR)
Aerodynamics
Wing Aspect Ratio
AR = b^2 / S
Aerodynamics
Coefficient of Drag
C_D = D / (q * S)
Aerodynamics
Coefficient of Lift
C_L = L / (q * S)
Aerodynamics
Dynamic Pressure
q = 1/2 * ρ * V^2
Aerodynamics
Mach Number
M = V / a
Propulsion
Specific Impulse
I_sp = F / (ṁ·g₀)
Propulsion
Rocket Thrust Equation
F = ṁv_e + (p_e − p_0)A_e
Aerodynamics
Drag Equation
D = ½ρv²C_D·A
Aerodynamics
Lift Equation
L = ½ρv²C_L·A