Vector Resolution - xy Plane - Airspeed/Groundspeed

[JeffNYC]

Homework Statement

Plane flies at constant ground speed of 400mph due East. There is a 50mph wind from the NW. Find airspeed and compass direction that will allow the plane to maintain its groundspeed of 400mph and due east direction.

Homework Equations

see b3

The Attempt at a Solution

Attached image has my attempt at the problem. I found the velocity vectors (I think correctly) of the Plane and the Wind, added the 2 vectors and found the resultant vector. How should I interpret this resultant vector (is it airspeed?) and how do I find the required compass direction?

Thank you,

Jeff

 

[edziura]

Your groundspeed vector is correct. The wind vector on your diagram is correctly oriented, but your component equation is incorrect; it would be if the wind were blowing in the opposite direction. It should be (50 cos 45)i - (50 sin 45)j.

The manner in which these terms are usually used is such that ground speed = airspeed + wind speed. The word "speed" really should be "velocity" as I suspect you realize. Given the information you have, airspeed = ground speed - windspeed, and so your equation relating these three vectors is not correct.

Basically what the question is asking is one which every pilot has to answer everytime he or she goes flying: If I want my velocity over the ground to be 400 mph E, and I have a wind behind and from the left (from the NW), at what velocity do I need fly the airplane (through the air) to achieve this. Qualitatively, the answer is less than 400mph since I have a bit of a tailwind pushing me eastward, and somewhat north of east to compensate for the wind tending to push me south. Ofcourse, GPS and computers have eliminated the need to any calculations "by hand".

 

[baba89]

Hello Jeff,

Your approach to the problem is correct. The resultant vector represents the airspeed of the plane. To find the required compass direction, you can use trigonometry to calculate the angle between the resultant vector and the due East direction. This angle will give you the direction in which the plane needs to fly in order to maintain its groundspeed of 400mph and due East direction.

I hope this helps! Keep up the good work in your studies of vector resolution. It is a very useful tool in understanding motion and forces. Good luck!