An aircraft flying under the guidance of a nondirectional beacon (a fixed radio

Question

An aircraft flying under the guidance of a nondirectional beacon (a fixed radio transmitter, abbreviated NDB) moves so that its longitudinal axis always points toward the beacon. A pilot sets out toward an NDB from a point at which the wind is at right angles to the initial direction of the aircraft; the wind maintains this direction. Assume that the wind speed and the speed of the aircraft through the air (its

An aircraft flying under the guidance of a nondirectional beacon (a fixed radio transmitter, abbreviated NDB) moves so that its longitudinal axis always points toward the beacon. A pilot sets out toward an NDB from a point at which the wind is at right angles to the initial direction of the aircraft; the wind maintains this direction. Assume that the wind speed and the speed of the aircraft through the air (its "airspeed") remain constant. (Keep in mind that the latter is different from the aircraft's speed with respect to the ground.) Locate the flight in the xy-plane, placing the start of the trip at (2,0) and the destination at (0,0) . Set up the differential equation describing the aircraft's path over the ground. For the question, the differential equation I think is :

Explanation / Answer

Given

y(x) = xsinh((-(wlog(x)) + c)/v)

substituting y=0 , x=2

sinhk=0 k=(-wlnx+c)/v

sinhk=(e^k-e^-k)/2 = 0

k=0

c=wln2

y(x) = xsinh((-wlnx+wln2)/v)

y(x) = xsinh(wln(2/x)/v)

which is already in explicit form

y = xsinh(wln(2/x)/v) (it is shown in terms of hyperbolic function as sinh(X) is hyperbolic)

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