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24 votes
24 votes
Diane starts to get frustrated that the plane is off-course, so she takes the controls. Shewants to get the plane to fly at an angle of 25° North of East, with a resultantgroundspeed of 140 kilometers per hour. She knows she must "aim" the airplane at anangle that compensates for wind. At what angle should she direct the airplane to fly?

User Sachin Kulkarni
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1 Answer

29 votes
29 votes

We know that the wind is blowing from north to south with a speed of 32 km per hour, let's call the wind vector W, then it can be written as


\vec{W}=-32\hat{j}

We also know that Diane wants the plane to fly north of east at an angle of 25° and a speed of 140 km/h. This means that the resultant should be


\vec{R}=140\cos 24\hat{i}+140\sin 24\hat{j}

Now, we would like to know at what angle Diane should fly tha plane. To find this we will introduce a vector T whose magnitud and direction are unknown.

Then


\vec{T}=T\cos \theta\hat{i}+T\sin \hat{j}

Now, we know that the resultant is


\vec{R}=\vec{T}+\vec{W}

then


\begin{gathered} 140\cos 25\hat{i}+140\sin 25\hat{j}=T\cos \theta\hat{i}+T\sin \theta\hat{j}+(-32\hat{j}) \\ 140\cos 25\hat{i}+140\sin 25\hat{j}=T\cos \theta\hat{i}+(T\sin \theta-32)\hat{j} \end{gathered}

Since the unit vector i and j are independent this gives us two equations


\begin{gathered} 140\cos 25=T\cos \theta \\ 140\sin 25=T\sin \theta-32 \end{gathered}

From the first equation we have that


T=(140\cos 25)/(\cos \theta)

Plugging the value of T in the second equation we have


140\sin 25=((140\cos25)/(\cos\theta))\sin \theta-32

Now we need to solve this equation for theta.


\begin{gathered} 140\sin 25=((140\cos25)/(\cos\theta))\sin \theta-32 \\ 140\sin 25+32=(140\cos 25)(\sin \theta)/(\cos \theta) \\ (\sin\theta)/(\cos\theta)=(140\sin 25+32)/(140\cos 25) \end{gathered}

Now we have to remember that


(\sin\theta)/(\cos\theta)=\tan \theta

hence


\begin{gathered} \tan \theta=(145\sin 25+32)/(145\cos 25) \\ \theta=\tan ^(-1)((145\sin 25+32)/(145\cos 25)) \\ \theta=35.7 \end{gathered}

Therefore Diane has to direct the airplane at an angle of 35.7° North of east.

User Arup Rakshit
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2.8k points