What heading and airspeed should you choose to reach your destination in time in the following? You are piloting a small plane, & you want to reach an airport 450km due South in 3.0 hrs. A wind is blowing from the west at 50km/h.
I know this is about vectors, I have drawn an illustration, but am still stuck about how to solve this problem.
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The situation is shown in the above figure. Orange dot represent the initial position of the plane. It is straightly on top of the final destination. If the plane head to the south then when it comes 450km to the south it will end up at a point as shown by the black dot because of the wind blowing from west will carry the plane to the east direction. So to stop at the final destination the plane should start its run towards a direction shown by the red dot.
Lets take angle between final destination of plane, Starting point and red dot as `alpha` . If we find `alpha` that is the direction that plane should go.
The plane should travel a distance of 450kn in 3 hrs if it travels without wind.
So the velocity of plane without wind = 450/3 = 150km/h
So if the velocity of the plane once its directed at a `alpha` angle is V;
`Vcosalpha = 150(km)/h`
But Vsinalpha = 50km/h since the plane travels under wind.
`Vcosalpha = 150` ---(1)
`Vsinalpha = 50` ----(2)
`tanalpha = 50/150 = 1/3`
`alpha = tan^(-1) (1/3) = 18.435deg`
From (1); V = 150/cos(18.435) = 158.11km/h
So the plane should direct 18.35deg from south to west at a speed of 158.11km/h to reach the final destination with in 3 hours.
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