A sailboat under auxiliary power is proceeding on a bearing 25° north of west at 6,25km/h in still water.Then a tail wind blowing 15km/h in the direction 35° south of west, alters the course of the sailboat.What is the resultant speed and direction of the saiboat (correct to 2 decimal places)?
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Expert's answer
2020-10-17T16:33:39-0400
Calculuating the components of vectorA1A2,A3A2andA3A1in the system of rectangular axesandA2is a point6.25units away fromAat a bearing of25°,A3is a point15units away fromAat a bearing of215°We first construct a diagram includingpointA1,A2,A3at the given bearingand determine the magnitude ofA3A2.Since the magnitude of velocityis speed (see the figure).Hence, thexandycomponents of vectorA1A2,andA3A1are respectively given byA1A2x=−6.25cos(25°)=−5.66A1A2y=6sin(25°)=2.64A1A2can be written asA1A2=A1A2xi+A1A2yj,whereiandjare unit vectors.∴A1A2=−5.66i+2.64jSimilarly,A3A1x=15cos(215°)=−12.29A3A1y=15sin(215°)=−8.60A3A1=−12.29i−8.60jBy the vector law of addition,A3A1+A1A2=A3A2∴A3A2=−17.95i−5.96j∣A3A2∣is the resultant speed anddirection of the sail boat.∴∣A3A2∣=17.952+5.962=18.91Therefore, the resultant speed is18.91km/h,to2d.p.Also, the direction of the sail boat is given byθ=arctan(17.955.96)=18.37°∴ The direction of the sail boat isW18.37°S
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