An aircraft flies along a straight trajectory given by the vector equation r(t) = (100 + 40t, 200 + 30t), where position is in kilometers and time t is in hours. What is the ground speed of the aircraft?
Matematica Aeronautica
Question · Easy
Two lines, one point: an emo ballad
Short
Solve for x in the exponential equation 4^(x + 1) = 32.
What is the angle between the heading vector u = (1, 1) and vector v = (0, 2)?
Question · Medium
What is the derivative of the altitude descent function h(t) = 10,000 - 50t - 2t² with respect to time t?
Solve the trigonometric equation 2·sin(x) - √3 = 0 for 0° ≤ x ≤ 180°.
Density altitude, explained
Evaluate the limit: lim (x → 0) [ sin(4x) / (2x) ].
An aircraft fuel consumption rate in liters per hour decreases in arithmetic progression over 5 cruise intervals: 1200, 1150, 1100, 1050, 1000. What is the total fuel consumed across these 5 intervals?
What is the equation of the circle representing an air defense radar coverage zone centered at (4, -2) with a radius of 10 km?
What is the maximum value of the function f(x) = -2x² + 8x + 3?
Solve the logarithmic inequality log₃(2x - 1) < 2.
In a 3D airspace coordinate system, what is the distance between the radar station at O(0, 0, 0) and an aircraft at P(3, 4, 12) km?
What is the integral ∫ (4x³ - 6x + 5) dx?
If matrix A = [[2, 3], [1, 4]], what is the trace of matrix A (tr(A))?
Simplify the expression cos(α - β) when α = 90° and β is any angle.
What is the slope of the line perpendicular to the runway centerline defined by the equation 3x - 6y + 7 = 0?
Find the roots of the cubic polynomial P(x) = x³ - 6x² + 11x - 6 = 0.
What is the equation of the tangent line to the curve y = x² at the point P(3, 9)?
How many different ways can an Air Force formation of 4 distinct aircraft (Alpha, Bravo, Charlie, Delta) be arranged in a line on the flight line?
What is the value of the expression log₁₀(25) + log₁₀(4)?
In a right triangle representing a climb profile, the hypotenuse is 10 km and the altitude gained is 6 km. What is the horizontal ground distance covered?
What is the area of a triangle formed by vectors u = (4, 0) and v = (0, 6) in the coordinate plane?
Find the inflection point of the cubic curve y = 2x³ - 12x² + 5x - 1.
Question · Hard
Solve the inequality (x - 1)/(x + 3) ≤ 0.
What is the general term of the sequence: 2, 5, 10, 17, 26, ...?
Convert an angle of 5π/6 radians into degrees.
Calculate the determinant of the 3×3 matrix with rows [1, 0, 0], [0, 4, 0], [0, 0, 5].
What is the value of the limit lim (x → ∞) [ (3x² + 5x - 2) / (6x² - x + 4) ]?
If vector A = (2, -1, 3) and vector B = (1, 4, -2), what is the sum vector A + B?
What is the derivative of f(x) = ln(3x² + 1)?
In a geometric progression, the 2nd term is 6 and the 5th term is 48. What is the common ratio r?
What is the domain of the function g(x) = ln(4 - x²)?
A fighter jet pulls out of a dive along a circular path of radius R = 500 meters at a constant speed of 100 m/s. What is its centripetal acceleration in m/s²?
Fisica E Aerodinamica
What is the value of the sum ∑ (from k=1 to 100) k?
Solve the system of equations: 2x - 3y = 7 and 4x + y = 7. What is the value of x + y?
What is the equation of the directrix of the parabola y² = 12x?
A transport aircraft of mass 20,000 kg takes off from a runway accelerating from rest to 80 m/s over a distance of 1,600 meters. What is the magnitude of the constant net force acting on the aircraft?
According to the aerodynamic lift equation L = 0.5 · C_L · ρ · v² · S, what happens to aerodynamic lift if true airspeed v is doubled while all other parameters remain constant?
What is the Mach number of an aircraft flying at 680 m/s in high-altitude air where the local speed of sound is 340 m/s?
A jet engine produces a constant thrust of 50,000 N while cruising at 250 m/s. What power is generated by the propulsion system?
According to the International Standard Atmosphere (ISA) model, what is standard sea-level atmospheric pressure and temperature?
What is the speed of sound in air as a function of absolute temperature T (where γ = 1.4 and R_specific = 287 J/(kg·K))?
A missile of mass 250 kg is launched vertically with an initial velocity of 300 m/s. Neglecting air resistance, what maximum altitude does it reach? (Use g = 10 m/s²)
What is the primary physical phenomenon that causes induced drag (resistência induzida) on an aircraft wing in flight?
In a level, unaccelerated turn at a bank angle of θ = 60°, what is the load factor (fator de carga, n) experienced by the airframe?
What is the total capacitance of two capacitors of 40 µF and 60 µF connected in series in an avionics power supply?
An aircraft altimeter operates on the principle of measuring which physical quantity?
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