After ﬂying for 15 min in a wind blowing 42 km/h at an angle of 20° south of east, an airplane pilot is over a town that is 55

Question

After ﬂying for 15 min in a wind blowing 42 km/h at an angle
of 20&deg; south of east, an airplane pilot is over a town that is 55 km
due north of the starting point. What is the speed of the airplane
relative to the air?

wolayemi · Answer

First off, the answer is 58.15 m/s. What youll do is is find out what distance the plane would have traveled if it only had he wind acting upon it. Then find out what speed it would have had to be going to end up in its actual spot (55km due north of the starting position).
They tell you that the wind is blowing at 42 km/h so we need to convert that to m/s: (42km/h)(1000m/1km)(1h/3600s) = 11.67 m/s. Now find the horizontal and vertical components of the wind speed; use SOHCAHTOA: vx = 11.67cos(20) = 10.97 m/s. vy = 11.67sin(20) = 3.99 m/s. 
Now what distance did the plane travel due to just the wind: d=vt. Again, break it into components: dx=vxt and dy=vyt. (10.97m/s)(900s) = 9873m and (3.99m/s)(900s) = 3591m.
After having moved 9873m in the horizontal direction and 3591m in the vertical direction from the starting point, how do you need to move to be 55000m due North of the starting point? Well we need to move an addditional 51409m north (55000m-3591m) and 9873m west. We already know the time, so we just need to use the equation v=x/t to solve for the component speeds, then combine them into one vector speed.
(9873m)/(900s) = 10.91 m/s  and (51409m)/(900s) = 57.12 m/s
Finally, combine them the the equation c=SQRT((A^2)+(B^2)), where 10.91 and 57.12 are A and B respectively. This is just the Pythagorean Theorem rearranged, btw. 
SQRT((10.91^2)+(57.12^2)) = 58.15 m/s
BONUS If you wanted direction of the plane too, use SOHCAHTOA just like earlier in the problem. tan&theta; = 57.12/10.91. The plane is moving ~10.81 degrees west of north.

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After ﬂying for 15 min in a wind blowing 42 km/h at an angle of 20° south of east, an airplane pilot is over a town that is 55

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