Answer to Question #175351 in Physics for Welriel

Question

A sailboat sails from Marblehead Harbor directly east for 45 nautical miles, then 60 south

of east for 20 nautical miles, returns to an easterly heading for 30 nautical miles, and sails

30 east of north for 10 nautical miles, then west for 62 nautical miles. At that time the boat

becomes becalmed and the auxiliary engine fails to start. The crew decides to notify the

Coast Guard of their position. What is their position?

Answer 1

Make a drawing (NOT in scale):

Displacement toward E:

$$X_1=45\text{ NM},\\ X_2=20\cos60°=10\text{ NM},\\ X_3=30\text{ NM},\\ X_4=10\cos60°=5\text{ NM},\\ X_5=-62\text{ NM},\\ X=45+10+30+5-62=8\text{ NM}.$$

Displacement toward S:

$$Y_1=0\text{ NM},\\ Y_2=20\cos30°=17.3\text{ NM},\\ Y_3=0\text{ NM},\\ Y_4=-10\cos30°=8.66\text{ NM},\\ Y_5=0\text{ NM},\\Y=0+17.3+0+8.66+0=25.96\text{ NM}.$$

Angle:

$$\theta=\arctan\frac{Y}{X}=72.9°\text{ S of E}.$$

Displacement from the starting point:

$$D=\sqrt{X^2+Y^2}=27.16\text{ NM}.$$