GATE Aerospace Engineering (AE) Calculators
Calculators for the numerical questions in GATE Aerospace Engineering, from the standard atmosphere and aircraft performance to compressible flow, structures and propulsion, each showing the working.
55 calculators · 6 converters · common conversions
Engineering Mathematics
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Eigenvalue Calculator
det(A − λI) = 0
Σλ = trace A, Πλ = det A
Example: The matrix [[4, 1], [2, 3]]: Eigenvalues: λ₁ = 5, λ₂ = 2.
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Matrix Rank Calculator
rank = number of non-zero rows in echelon form
unique if rank A = rank [A|b] = n; none if rank A < rank [A|b]
Example: The matrix [[1, 2, 3], [2, 4, 6], [1, 0, 1]]: Rank = 2.
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- Bisection Method Calculator – f(x) and an interval where it changes sign → root by bisection or false position, with every iteration.
- Newton-Raphson Method Calculator – f(x), optional f′(x) and a starting guess → root, iteration table and convergence check.
- Numerical Integration Calculator – f(x), limits and intervals → trapezoidal, Simpson 1/3 and 3/8 results with a reference value.
- Numerical Differentiation Calculator – f(x), x and step h → forward, backward and central difference derivatives with their errors.
- Lagrange Interpolation Calculator – Data points and x → interpolated value, Lagrange basis values and the polynomial.
- Complex Number Calculator – Two complex numbers and an operation → result, modulus, argument and conjugate.
- Bayes’ Theorem Calculator – Priors and likelihoods → total probability and posterior probabilities, with a Bayes table.
- Normal Distribution Calculator – Mean, standard deviation and bounds → normal probabilities, z-scores, density and percentiles.
Flight Mechanics
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Standard Atmosphere Calculator
T = 288.15 − 0.0065h, p = 101325 (T/288.15)^5.2559
troposphere to 11 km; isothermal 216.65 K from 11 to 20 km; ρ = p/(RT), a = √(γRT)
Example: The standard atmosphere at 5000 m: Temperature T = 255.65 K.
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Airspeed Calculator
EAS = TAS √(ρ/ρ₀), q_c = p[(1 + 0.2M²)^3.5 − 1]
CAS = a₀√(5[(q_c/p₀ + 1)^(2/7) − 1]); a₀ = 340.294 m/s, p₀ = 101325 Pa
Example: 250 knots calibrated airspeed at 10,000 ft pressure altitude: True airspeed TAS = 288.702 kn.
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- Lift Equation Calculator – Density, airspeed, wing area and C_L → lift; or solve for any one of them.
- Drag Polar Calculator – C_D0, aspect ratio, Oswald efficiency and C_L → C_D, L/D and the optimum lift coefficients.
- Stall Speed Calculator – Mass, wing area, C_L,max and density → stall speed, manoeuvring speed and reference speeds.
- Take-off Distance Calculator – Mass, wing area, C_L,max, thrust and density → take-off ground roll.
- Rate of Climb Calculator – Thrust, drag, airspeed and mass → rate of climb, climb angle and excess power.
- Glide Ratio Calculator – L/D and height (and speed) → glide distance, glide angle, sink rate and time.
- Breguet Range Calculator – L/D, fuel consumption, speed or propeller efficiency and weights → range and endurance.
- Turn Performance Calculator – Airspeed and bank angle → turn radius, rate, load factor and time for a full circle.
Space Dynamics
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Hohmann Transfer Calculator
Δv₁ = √(μ/r₁)(√(2r₂/(r₁ + r₂)) − 1)
Δv₂ = √(μ/r₂)(1 − √(2r₁/(r₁ + r₂))); t = π√(a³/μ), a = (r₁ + r₂)/2
Example: From a 300 km circular orbit to geostationary altitude (35,786 km): Total Δv = 3.89523 km/s.
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Elliptical Orbit Calculator
T = 2π√(a³/μ), v = √(μ(2/r − 1/a))
a = (r_p + r_a)/2, e = (r_a − r_p)/(r_a + r_p), energy = −μ/(2a)
Example: An Earth orbit with a 400 km perigee and a 4000 km apogee: Orbital period T = 131.615 min.
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- Orbital Velocity Calculator – A body and an altitude → orbital speed, period and orbit radius.
- Escape Velocity Calculator – A planet (or any mass and radius) → escape velocity, surface orbit speed and surface gravity.
- Gravitational Force Calculator – Two masses and their separation → gravitational force; or solve for a mass or the distance.
Aerodynamics
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Mach Number Calculator
M = V / a, a = √(γRT)
γ = 1.4 and R = 287.05 J/(kg·K) for air; T in kelvin
Example: An aircraft at 250 m/s true airspeed at 11 km: Mach number M = 0.847258.
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Thin Airfoil Theory Calculator
c_l = 2π(α − α_L0), a = a₀/(1 + a₀/(πeAR))
Kutta–Joukowski L′ = ρV∞Γ, so Γ = ½V∞ c c_l
Example: A cambered airfoil (α_L0 = −2°) at 5° on a wing of aspect ratio 8: Section lift coefficient c_l = 0.767636.
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- Induced Drag Calculator – C_L and aspect ratio (or span and area) → induced drag coefficient and induced angle.
- Boundary Layer Calculator – Free-stream speed, distance and viscosity → boundary-layer thicknesses and skin friction.
- Pitot Tube Calculator – Pitot and static pressures → airspeed (low speed) or Mach number (subsonic and supersonic).
- Bernoulli Equation Calculator – Pressure, velocity and elevation at 1, velocity and elevation at 2 → pressure at 2 and heads.
- Reynolds Number Calculator – Velocity, diameter, density and viscosity in any units → Re and flow regime.
- Poiseuille Flow Calculator – Radius, length, viscosity and pressure difference → laminar flow rate; or the pressure or radius.
- Isentropic Flow Calculator – Mach number (or an area, pressure or temperature ratio) → the full isentropic table row.
- Normal Shock Calculator – Upstream Mach number → downstream Mach and every property ratio across a normal shock.
- Oblique Shock Calculator – Upstream Mach number and deflection angle → shock angle, M₂ and property ratios.
- Prandtl-Meyer Expansion Calculator – Mach number and turning angle → Mach after an expansion fan, or ν(M) and its inverse.
Structures
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Von Mises Stress Calculator
σ_v = √(½[(σ₁ − σ₂)² + (σ₂ − σ₃)² + (σ₃ − σ₁)²])
plane stress: σ_v = √(σx² − σxσy + σy² + 3τxy²); Tresca σ₁ − σ₃
Example: A plane stress state of σx = 100 MPa and τxy = 50 MPa: von Mises stress σ_v = 132.288 MPa.
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Mohr’s Circle Calculator
σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]
centre C = (σx + σy)/2, radius R = τmax
Example: σx = 80 MPa, σy = −40 MPa and τxy = 45 MPa: Major principal stress σ₁ = 95 MPa.
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- Thin-Walled Torsion Calculator – Torque and section geometry → shear flow, shear stress, torsion constant and angle of twist.
- Shaft Torsion Calculator – Torque, diameters, G and length → maximum shear stress, polar moment, angle of twist and power.
- Bending Stress Calculator – Bending moment and cross-section → maximum bending stress, I, y and section modulus Z.
- Beam Deflection Calculator – Cantilever or simply supported, point load or UDL → maximum deflection, slope, moment and shear.
- Euler Buckling Calculator – End conditions, E, I, length and area → Euler critical load, critical stress and slenderness.
- Strain Energy Calculator – Load type, load, length, area and E → maximum stress, extension and strain energy stored.
- Elastic Constants Calculator – Any two of E, G, K and ν → the other elastic constants and Lamé’s λ.
- Natural Frequency Calculator – Mass, stiffness and damping → natural frequency, damping ratio, damped frequency and log decrement.
- Transmissibility Calculator – Frequency ratio and damping ratio → transmissibility, magnification factor and isolation.
Propulsion
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Brayton Cycle Calculator
η = 1 − 1/rp^((γ−1)/γ)
rp pressure ratio; optimum rp = (T₃/T₁)^(γ/2(γ−1)) for maximum work
Example: rp = 10, T₁ = 300 K, T₃ = 1200 K, ideal compressor and turbine: Thermal efficiency η = 48.2053 %.
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Jet Engine Thrust Calculator
F = ṁₐ[(1 + f)vₑ − V₀] + Bṁₐ(v_f − V₀)
TSFC = ṁ_f/F; η_p = FV₀/ΔKE; η_th = ΔKE/(ṁ_f Q_R)
Example: A turbojet with 50 kg/s of air, f = 0.02, flying at 250 m/s with a 600 m/s jet: Thrust F = 18.1 kN.
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- Axial Compressor Stage Calculator – Blade speed, whirl velocities, inlet temperature and efficiency → work, temperature rise, pressure ratio and reaction.
- Nozzle Flow Calculator – Chamber pressure and temperature, throat area and expansion ratio → mass flow, exit conditions, thrust and Isp.
- Rocket Thrust Calculator – Mass flow, exhaust velocity, exit and ambient pressure and exit area → thrust and specific impulse.
- Rocket Equation Calculator – Specific impulse and mass ratio → Δv; or the mass, propellant or Isp needed for a Δv.
- Air–Fuel Ratio Calculator – Fuel formula and excess air → stoichiometric and actual air–fuel ratio, φ and λ.
Unit Converters for GATE Aerospace (AE)
- Speed Converter – km/h, mph, m/s, ft/s, knots
- Pressure Converter – Pa, kPa, psi, bar, inH₂O, mmHg, torr, atm
- Temperature Converter – °C, °F, K, °R
- Density Converter – kg/m³, g/cm³, g/mL, lb/ft³, lb/gal, mg/L
- Force Converter – N, kN, lbf, kgf, kip, dyn, ozf, gf
- Angle Converter – degrees, radians, gradians, arcminutes
Common GATE Aerospace (AE) Conversions
- mile per hour to kilometer per hour
- kilometer per hour to mile per hour
- pascal to psi
- psi to pascal
- Celsius to Fahrenheit
- Fahrenheit to Celsius
- gram per cubic centimeter to kilogram per cubic meter
- kilogram per cubic meter to gram per cubic centimeter
- newton to pound-force
- pound-force to newton
- radian to degree
- degree to radian
How to use these GATE Aerospace Engineering calculators
GATE AE numerical questions lean on a handful of relations used again and again: the standard atmosphere, the lift equation and drag polar, the performance equations for stall, turning, climbing, gliding and range, Kepler’s laws, the isentropic and shock relations of compressible flow, thin-walled structures and the thrust equations of jets and rockets. Each has its own calculator here, alongside the structures and vibration tools shared with mechanical engineering.
The atmosphere follows ICAO Doc 7488 with R = 287.05287 J/(kg·K), the compressible-flow tools reproduce the NACA 1135 tables for any ratio of specific heats, and orbital calculations use measured gravitational parameters GM rather than G times a mass. Every input takes the unit you have, so knots, feet, kilonewtons and bar can go straight in.
Work each question by hand, then check it here. When answers differ, compare the steps: the usual culprits are degrees versus radians, altitude versus radius from the planet’s centre, and static versus stagnation quantities.