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.)

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