Chapter 6.3, Problem 109E

To determine

To find: The direction and speed of the wind.

Answer to Problem 109E

  $130{\text{ km/h and 207}}^{\circ }$

Explanation of Solution

Given: An airplane is flying in the direction ${148}^{\circ }$ with an airspeed of 860 kilometers per hour. Because of the wind, its groundspeed and direction are 800 kilometers per hour and ${140}^{\circ }$ , respectively.

  

First draw component of speed of plane and speed of plane w.r.t to wind as shown in diagram.

  

First write airplane speed in form of component.

  $\begin{array}{l}{V}_P=860\cos {58}^{\circ }i-860\sin {58}^{\circ }j\\ V_P=455.73i-729.32j\end{array}$

Now, write the speed of plane w.r.t wind in form of component.

  $\begin{array}{l}{V}_{PW}=800\cos {50}^{\circ }i-800\sin {50}^{\circ }j\\ V_{PW}=514.23i-612.84j\end{array}$

Now, find the speed of wind using above velocity.

Let velocity of wind be $V_W$

  $\begin{array}{l}{V}_{PW}=V_P-V_W\\ V_W=V_P-V_{PW}\\ V_W=\left(455.73i-729.32j\right)-\left(514.23i-612.84j\right)\\ V_W=\left(455.73-514.23\right)i-\left(729.32-612.84\right)j\\ V_w=-58.50i-116.49j\end{array}$

Speed of wind, $\left|V_W\right|=\sqrt{{\left(-58.50\right)}^2+{\left(-116.49\right)}^2}\approx 130$

Direction of wind,

  $\begin{array}{l}\theta ={\tan }^{-1}\left(\frac{-116.49}{-58.50}\right)\\ \theta \approx {207}^{\circ }\end{array}$

Hence, the speed of wind is 130 km/h and direction ${207}^{\circ }$ .