Question

The sailboat sails to the northeast at an angle of 65 ° to the east. After 1.2 km and a short stop, the sailboat changes direction and continues to sail to the southeast at an angle of 15 ° to the east, making a shift of 1.8 km. Find the total displacement of the sailboat.

Answer 1

Let's find $x$ and $y$ components of the total displacement of the sailboat:

$$d_x=1.2\ km\cdot cos65^{\circ}+1.8\ km\cdot cos(360^{\circ}-15^{\circ})=2.24\ km,$$ $$d_y=1.2\ km\cdot sin65^{\circ}+1.8\ km\cdot sin(360^{\circ}-15^{\circ})=0.62\ km.$$

Then, the total displacement can be found from the Pythagorean theorem:

$$d=\sqrt{d_x^2+d_y^2}=\sqrt{(2.24\ km)^2+(0.62\ km)^2}=2.32\ km.$$

We can find the angle as follows:

$$\theta=sin^{-1}(\dfrac{d_y}{d})=sin^{-1}(\dfrac{0.62\ km}{2.32\ km})=15.5^{\circ}.$$

The sign plus means that total displacement has direction $15.5^{\circ} N\ of\ E$.

Therefore, the total displacement $d$ has magnitude 2.32 km and direction of $15.5^{\circ} N\ of\ E$.