The airplane m is heading due north. Calculate the acceleration of the airplane

Question

The airplane m is heading due north. Calculate the acceleration of the airplane measured by the person sitting at the Earth center without rotation. Use the following configuration (ijk-frame is an airplane-fixed frame and IJK is ECIF) and write the answer using 'jk expression by showing the following steps: i. Define an absolute position vector Rn in ijk-frame. ii Find the absolute acceleration vector ink-frane by taking the time derivative of R n ink- frame. Airplane Earth Surface Figure 1: Airplane on Eath

Explanation / Answer

define i, j, k frame as fixed to hte body of th eplane

I,J,K frame fixed to the center of the earth

alos

the earth is non rotating

the plane is moving towards north at an angle phi to the I axis

altitude = h

radius of earth = Re

from diagram we can see

i = J

I = kcos(phi) - jsin(phi)

K = ksin(phi) + jcos(phi)

a. hence

absolute position vector of the airplane

Rm = (R + h)cos(phi)I + (R + h)sin(phi)K

substituting I and K

Rm = (R + h)k

b. acceleration vector can be found as

d^Rm/dt^2 = d^2h/dt^2 k

where d^2h/dt^2 is rate of change of vertical speed of the airplane

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