GATE AE flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Percentage increase formula

A value rising from A to B increases by (B − A)/A × 100 percent; a fall uses the same denominator A, not B.

Combined work rate

If two agents complete a job in x and y days separately, their combined one-day rate is 1/x + 1/y, so joint time is xy/(x+y).

Permutation with repetition

Arrangements of n objects with n1 identical of one kind and n2 of another count n!/(n1! n2!); divide by each identical group's factorial.

Categorical syllogism rule

From 'All A are B' one cannot infer 'All B are A'; from 'Some A are B' one can infer 'Some B are A'.

Eigenvalue trace relation

For an n×n matrix, the sum of eigenvalues equals its trace and their product equals its determinant; repeated eigenvalues count with algebraic multiplicity.

Rank-nullity theorem

For an m×n matrix, rank plus nullity equals n; nullity is the dimension of the solution space of Ax = 0.

Laplace derivative rule

L{f′(t)} = sF(s) − f(0); the transform of a derivative multiplies by s and subtracts the initial value.

Trapezoidal rule error order

Composite trapezoidal rule error is O(h²); composite Simpson's rule is O(h⁴) and requires an even number of subintervals.

Load factor definition

n = L/W = 1/cos φ in a level coordinated turn; structural limits bound allowable n and hence turn rate and stall speed.

Stall speed relation

V_stall = sqrt(2W/(ρ S C_Lmax)); increasing altitude lowers density and raises true stall speed for the same weight and C_Lmax.

Breguet range equation

For a jet, range = (V/c_T)(L/D) ln(W_initial/W_final); propeller aircraft use an analogous form with propeller efficiency and brake specific fuel consumption.

Static margin

Static margin = (x_NP − x_CG)/MAC; a positive value places the CG ahead of the neutral point and provides pitch stiffness.

Vis-viva equation

v² = μ(2/r − 1/a) gives orbital speed at radius r for a conic with semi-major axis a; a is negative for hyperbolas.

Escape velocity

v_escape = sqrt(2μ/r) at radius r; it equals sqrt(2) times the circular orbital speed at the same radius.

Hohmann transfer delta-v

A two-impulse transfer between coplanar circular orbits uses a periapsis burn and an apoapsis burn; its total delta-v is the sum of their magnitudes.

Kepler third law

T² = 4π²a³/μ for a two-body orbit; period depends only on semi-major axis and gravitational parameter, not eccentricity.

Routh-Hurwitz criterion

For a polynomial characteristic equation, all first-column entries of the Routh array must be positive; sign changes equal right-half-plane poles.

Gain margin definition

GM = 1/|G(jω_pc)| at the phase crossover where phase is −180°; in decibels GM_dB = −20 log10|G(jω_pc)|.

Phase margin definition

PM = 180° + ∠G(jω_gc) at gain crossover where |G(jω_gc)|=1; positive PM indicates a stable closed-loop system with relative stability.

Controllability matrix

For xdot = Ax + Bu, form C = [B AB ... A^{n−1}B]; full rank n means every state is controllable.

Reynolds number

Re = ρVL/μ = VL/ν; it measures inertial to viscous forces and determines laminar, transitional, or turbulent boundary-layer behavior.

Kutta condition

At a sharp trailing edge, flow leaves smoothly with finite velocity; this condition fixes airfoil circulation and hence lift in inviscid thin-airfoil theory.

Thin airfoil lift slope

For a thin airfoil in incompressible flow, the lift-curve slope is 2π per radian; a finite aspect ratio reduces the wing lift slope.

Prandtl-Glauert correction

In subsonic compressible flow, pressure and lift coefficients scale by 1/sqrt(1−M∞²); the correction is invalid near sonic conditions.

Euler buckling load

P_cr = π²EI/(KL)² for a slender column; K depends on end conditions; high slenderness causes buckling before yield.

Bending stress formula

σ = −My/I gives normal stress at distance y from the neutral axis; maximum magnitude occurs at the outer fiber with largest |y|.

Torsion shear stress

For a circular shaft, τ = Tr/J, maximum at the outer radius; J is the polar second moment of area.

Von Mises yield criterion

Yielding occurs when von Mises stress reaches yield strength; for plane stress σ_vm = sqrt(σx² − σxσy + σy² + 3τxy²).

Rocket thrust equation

F = mdot v_e + (p_e − p_a)A_e; momentum and pressure terms contribute, and vacuum thrust exceeds sea-level thrust.

Specific impulse

Isp = thrust/(mdot g0), expressed in seconds when g0 is standard gravity; higher Isp means less propellant mass for a given impulse.

Brayton cycle efficiency

Ideal Brayton thermal efficiency depends on pressure ratio: η = 1 − 1/r_p^{(γ−1)/γ}, assuming isentropic compression and expansion.

Choked nozzle condition

A convergent nozzle chokes when exit pressure reaches critical pressure p*/p0 = (2/(γ+1))^{γ/(γ−1)}; mass flow then depends on throat area and chamber conditions.