CATAGORY: Mechanics
QUESTION: I think this question is Kinematics...
A jet pilot sets his compass due east. After 30 min, during which he
maintained a constant speed of 400km/h with repect to the air, he finds
himself flying over a city located 220 km east and 20km south of his
point of departure, Determine the wind velocity in magnitude and
direction.

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From the figure 1. below we see that in the X direction (east) the plane
travels
220km in 0.5hours (30min) and -20km in the Y direction (south) .  We will 
apply the basic formula relating position to time for a constant velocity.

(1)  d = vt

    distance = velocity X time.
     
This is a vector formula so we can break this up into X and Y components
of motion to give

(2)  x = V(x)t
     y = V(y)t
     
where x = location to the east of the starting point

      y = position to the north of the starting position
          note:if y is negative then the position is south
          of the starting position.
          
       V(x) = east component of velocity
       
       V(y) = north component of velocity (see note on y)

Now the plane flies due east at 400km/hr with respect to the air but is
also being carried aling by the wind so we have 
       
(3)      V(x) = 400km/hr + V(x,wind)
         V(y) = 0km/hr   + V(y,wind)

where V(x,wind) and V(y,wind) are the x and y coordinates of the wind with 
respect to the ground. Substituting these into (2) gives

(4)  x = (400km/hr + V(x,wind))t
     y = V(y,wind)t
     
now lets substitute some numbers into (4). From figure 1.

     x = 220km
     y = -20km
     t = 0.5hr
     
     so
     
(5)  220km = (400km/hr + V(x,wind)) X 0.5hr
     -20km = V(y,wind) X 0.5hr         


figure 1.

  \                220km
>-->- ....................................           ^ y
  /                                      .           |
                                         .           |
                      \                  .           |
                    >-->-                . 20km      -------> x
                      /                  .
                                         .\
                                        >-->- 
                                          /
                      

Solving (5) for V(x,wind) and V(y,wind) gives

(6)  V(x,wind) = 220km/0.5hr -400km/hr = 40km/hr
     V(y,wind) = -20km/0.5hr = -40km/hr
 
      *****************************
      *                           *     
      *    V(x,wind) =  40km/hr   *
      *    V(y,wind) = -40km/hr   *   
      *                           *
      *****************************
      
This means the wind is comming out of the north-west and blowing toward
the south-east.  The magnitude of the wind velocity is



       |V| = sqrt( (40km/hr)^2 + (-40km/hr)^2) = 56.6 km/hr
       
      ************************
      *                      *     
      *   |V| = 56.6km/hr    *       
      *                      *
      ************************       
(note: sqrt means square root and a^2 means a times a.)