Answer to Question #102632 in Electricity and Magnetism for Ali hamza

Question #102632
A commuter airplane takes the route shown in Figure 3.20. First, it flies from the origin of the coordinate system shown to city A, located 175 km in a direction 30.0° north of east. Next, it flies 153 km 20.0° west of north to city B. Finally, it flies 195 km due west to city C. Find the location of city C relative to the origin.
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Expert's answer
2020-02-10T09:27:33-0500

The route of  airplane look like one can see in figure.


OA=175 km, AB=153 km, BC=195 km

OA=(OAcos30°)i^+(OAsin30°)j^=175km32i^+175km12j^==151.55kmi^+87.5kmj^\vec{OA}=(OA\cdot cos30\degree)\cdot \hat {i}+(OA\cdot sin30\degree)\cdot \hat j=175km\cdot \frac{\sqrt{3}}{2} \hat i+ 175km\cdot \frac{1}{2} \hat j=\\= 151.55km\cdot \hat i+87.5km \cdot \hat j

AB=ABcos(90°+20°)i^+ABsin(90°+20°)j^==ABcos(110°)i^+ABsin(110°)j^=153km(0.342)i^+153km0.9397j^=52.3kmi^+143.8kmj^\vec{AB}=AB\cdot cos(90\degree+20\degree)\cdot \hat i+AB\cdot sin(90\degree+20\degree)\cdot \hat j=\\=AB\cdot cos(110\degree)\cdot \hat i+AB\cdot sin(110\degree)\cdot \hat j=153km\cdot (-0.342)\cdot \hat i+153km\cdot 0.9397\cdot \hat j=-52.3km\cdot \hat i +143.8km\cdot \hat j

BC=BCi^=195kmi^\vec {BC}=-BC\cdot \hat i=-195km\cdot \hat i

OC=OA+AB+BC\vec {OC}=\vec {OA}+\vec {AB}+\vec {BC}

OC=(151.5552.3195)kmi^+(87.5+143.8)kmj^=95.8kmi^+231.3kmj^\vec {OC}=(151.55-52.3-195)km\cdot \hat i+(87.5+143.8)km \cdot \hat j=-95.8km\cdot \hat i+231.3km \cdot \hat j

OC=(95.8km)2+(231.3km)2=250kmOC=\sqrt{(95.8km)^2+(231.3km)^2}=250km

α=sin1(95.8250)=22.5°\alpha=sin^{-1}(\frac{95.8}{250})=22.5\degree

Answer: City C located 250 km in a direction 22.5°22.5 \degree west of nord.


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