Answer to Question #94634 in Mechanics | Relativity for Ali hamza

Question #94634
A wind is blowing at 50 km h−1
from a direction 45° west of north. The pilot of an airplane wishes to fly on
a route due north from an airport. The airspeed of the airplane is 250 km h−1
.
(a)In what direction must the pilot point the nose of the airplane?
(b) What will be the airplane’s speed relative to the ground?
1
Expert's answer
2019-09-19T09:43:30-0400

Resulting aircraft speed:

v=u+w,\vec{v} = \vec{u} + \vec{w},

where u\vec{u} is the own aircraft airspeed, w\vec{w} is the wind speed.


Since wind is blowing in 45 degrees angle,

wy=wcos45°=wx=wsin45°.w_y = w\cos45\degree = w_x = w\sin45\degree.


Projecting the vectors on axes in absolute values:

  • in north(+)/south(-) axis: vy=uywy=ucosαwcos45°;v_y = u_y - w_y = u\cos\alpha - w\cos45\degree;
  • in east(+)/west(-) axis: 0=uxwx=usinαwsin45°.0 = u_x-w_x = u\sin\alpha - w\sin45\degree .

(here vx=0v_x=0 because we know the aircraft in result is heading due north).


From the 2nd equation,

sinα=wusin45°;α=arcsin(wusin45°)=arcsin(50kmh250kmh22)8.13°.\sin\alpha = \frac{w}{u}\sin45\degree; \\ \alpha = \arcsin(\frac{w}{u}\sin45\degree) = \arcsin(\frac{50\frac{\text{km}}{\text{h}}}{250\frac{\text{km}}{\text{h}}}\frac{\sqrt2}{2}) \approx 8.13\degree.

The above angle is the direction of pointing the nose of the aircraft, west of north.


From the 1st equation,

v=vy=u1sin2αwcos45°=u1(wusin45°)2wcos45°==250kmh1(50kmh250kmh22)250kmh22212.13kmh.v = v_y = u\sqrt{1-\sin^2\alpha} - w\cos45\degree = u\sqrt{1-(\frac{w}{u}\sin45\degree)^2} - w\cos45\degree = \\ = 250\frac{\text{km}}{\text{h}}\sqrt{1-(\frac{50\frac{\text{km}}{\text{h}}}{250\frac{\text{km}}{\text{h}}}\frac{\sqrt2}{2})^2} - 50\frac{\text{km}}{\text{h}}\frac{\sqrt2}{2} \approx 212.13 \frac{\text{km}}{\text{h}}.



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