Aerospace & Flight formulas — 25 solvers

Burn Time from Propellant Load

t_b = \frac{m_p}{\dot m}

Aerospace & FlightHow long an engine can fire: propellant on board divided by the rate it leaves at. Trivial arithmetic, and the number a whole stage is designed around — nozzle cooling, tank size and structural loads all follow from it.

Characteristic Velocity (c*)

c^{*} = \frac{p_c A_t}{\dot m}

Aerospace & FlightChamber pressure times throat area divided by mass flow. It measures the COMBUSTION alone — how much pressure the propellant makes per unit of flow — and is deliberately blind to whatever the nozzle does downstream of the throat.

Drag Equation from the Drag Coefficient

D = q \, S \, C_D

Aerospace & FlightThe drag on an aircraft written the way an aerodynamicist writes it: dynamic pressure times the SAME wing reference area used for lift, times the drag coefficient. Not the frontal area a car's drag is built on.

Drag Polar

C_D = C_{D0} + C_{Di}

Aerospace & FlightThe whole drag of an aircraft split in two: a part that is there whatever the wing is doing, and a part that is the price of lift. Every performance calculation an aircraft ever needs starts from this sum.

Equivalent Airspeed from True Airspeed

V_E = V \sqrt{\frac{\rho}{\rho_0}}

Aerospace & FlightThe speed the wing thinks it is doing: true airspeed scaled by the square root of the density ratio, so that ½ρV² and ½ρ₀V_E² come out the same. The bridge between the number on the panel and the number over the ground.

Glide Distance from Altitude

x = h \cdot \frac{L}{D}

Aerospace & FlightHow far an aircraft travels over the ground while descending a given height, in still air: altitude multiplied by the lift-to-drag ratio. The calculation every pilot does silently when the engine goes quiet.

Induced Drag Coefficient

C_{Di} = \frac{C_L^{2}}{\pi \, AR \, e}

Aerospace & FlightThe drag a wing cannot avoid paying for the lift it makes, from Prandtl's lifting-line theory: lift coefficient squared over π times aspect ratio times the span efficiency factor. It grows with the SQUARE of C_L, which is why it dominates at low speed.

Isentropic Pressure Ratio

\frac{p_0}{p} = \left(1 + \frac{\gamma - 1}{2} M^{2}\right)^{\frac{\gamma}{\gamma - 1}}

Aerospace & FlightThe pressure a moving gas would reach if brought smoothly to rest, over the pressure it has while moving. The same physics as the temperature ratio, raised to γ/(γ−1) — and the relation that sets a nozzle's exit pressure.

Isentropic Temperature Ratio

\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2} M^{2}

Aerospace & FlightHow much hotter a gas is when it is brought to rest than when it is moving: the stagnation temperature over the static temperature, set entirely by Mach number and the ratio of specific heats.

Lift Equation

L = q \, S \, C_L

Aerospace & FlightThe lift a wing produces: dynamic pressure times reference area times the lift coefficient. Three of the four terms are things you can measure with a tape and a gauge, and the fourth is where the whole of aerodynamics lives.

Lift-to-Drag Ratio

L/D = \frac{L}{D}

Aerospace & FlightLift divided by drag: how many newtons of weight an aircraft carries for every newton it has to push against. The single number that says how good a flying machine is, and the one a sailplane pilot lives by.

Nozzle Area Ratio and Mach Number

\frac{A}{A^{*}} = \frac{1}{M}\left[\frac{2}{\gamma+1}\left(1 + \frac{\gamma-1}{2}M^{2}\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}

Aerospace & FlightThe isentropic relation between how far a nozzle has opened out and how fast the gas is moving there. It is what turns a throat into supersonic exhaust, and the reason a rocket nozzle is a bell rather than a hole.

Propellant Mass Flow Rate from Thrust and Isp

\dot m = \frac{F}{I_{sp} \, g_0}

Aerospace & FlightHow much propellant an engine drinks per second to hold a given thrust at a given specific impulse. The bridge between the thrust a mission needs and the tankage it has to carry to get it.

Rate of Climb from Excess Power

RoC = \frac{P_a - P_r}{W}

Aerospace & FlightHow fast an aircraft climbs: the power left over after drag has been paid for, divided by the weight it has to lift. Climb is not a thrust question but a POWER question, and this is why.

Rocket Mass Ratio from Delta-v

MR = e^{\Delta v / v_e}

Aerospace & FlightThe ratio of loaded mass to burnout mass a vehicle needs in order to deliver a given delta-v. The rocket equation turned inside out, and the fastest way to see how brutally the exponential punishes an ambitious mission.

Rocket Thrust from Mass Flow

F = \dot m \, v_e

Aerospace & FlightThe momentum term of the thrust equation: mass leaving per second multiplied by the speed it leaves at. Newton's third law with a stopwatch on it, and the whole of the thrust when the nozzle is perfectly expanded.

Rocket Thrust with the Pressure Term

F = \dot m \, v_e + (p_e - p_a) A_e

Aerospace & FlightThe complete thrust equation: the momentum the exhaust carries away, plus whatever the gas pressure at the nozzle exit gains or loses against the ambient pressure outside. The reason an engine's thrust rises as it climbs.

Specific Impulse and Exhaust Velocity

I_{sp} = \frac{v_e}{g_0}

Aerospace & FlightSpecific impulse in seconds is the effective exhaust velocity divided by standard gravity — exactly 9.80665 m/s², a defined constant and not the local gravity wherever the engine happens to be firing.

Stall Speed

V_s = \sqrt{\frac{2W}{\rho \, S \, C_{L,max}}}

Aerospace & FlightThe slowest speed at which a wing can still carry the weight, reached when the lift coefficient is at its maximum. Everything an aircraft does slowly — approach, touchdown, minimum turn radius, runway length — is a multiple of this number.

Standard Atmosphere Density (Troposphere)

\rho = \rho_0 \left(1 - \frac{L h}{T_0}\right)^{\frac{g_0}{R L} - 1}

Aerospace & FlightAir density at altitude on a standard day, from the troposphere layer of the ICAO and U.S. Standard Atmosphere: a lapse rate of 6.5 K per kilometre from 15 °C at sea level, in hydrostatic balance. Valid to the tropopause at 11 km and no further.

Thrust-to-Weight Ratio

\frac{T}{W} = \frac{F}{m \, g}

Aerospace & FlightThrust divided by weight. Below 1 the vehicle cannot leave the pad at all; above 1, the excess is what accelerates it. This is the one place in rocketry where LOCAL gravity is the right number to use.

Total Delta-v Across Stages

\Delta v_{tot} = \Delta v_1 + \Delta v_2 + \Delta v_3

Aerospace & FlightDelta-v simply adds across stages, while the mass ratios that produce it multiply. That mismatch between a sum and a product is the entire argument for staging, and it is worth seeing written down.

Tsiolkovsky Rocket Equation

\Delta v = v_e \ln\!\left(\frac{m_0}{m_f}\right)

Aerospace & FlightThe velocity a rocket gains is its exhaust velocity times the natural logarithm of the ratio of its starting mass to its burnout mass. Derived by Konstantin Tsiolkovsky in 1903, and still the equation that decides whether a mission is possible.

Wing Aspect Ratio

AR = \frac{b^{2}}{S}

Aerospace & FlightHow slender a wing is: span squared divided by planform area. For a rectangular wing it is simply span over chord, and for every other wing it is the number that decides how expensive lift will be.

Wing Loading

w = \frac{W}{S}

Aerospace & FlightWeight carried per unit of wing area. One number, and it sets the stall speed, the ride in turbulence, the turn radius and the runway length — which is why it is the first figure quoted about any new design.