Question

You decide to go for a walk on the tundra to enjoy the Northern Lights. If you walk 2.0 km [W] and then 3.2 km [N 20 W] and then 1.5 km [S], what is your total displacement?

Answer 1

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

$$d_x=2.0\ km\cdot cos180^{\circ}+3.2\ km\cdot cos160^{\circ}=-5.0\ km,$$ $$d_y=3.2\ km\cdot sin160^{\circ}+1.5\ km\cdot sin270^{\circ}=-0.405\ km.$$

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

$$d=\sqrt{d_x^2+d_y^2}=\sqrt{(-5.0\ km)^2+(-0.405\ km)^2}=5.02\ km.$$

We can find the angle as follows:

$$\theta=sin^{-1}(\dfrac{d_y}{d})=sin^{-1}(\dfrac{-0.405\ km}{5.02\ km})=-4.63^{\circ}.$$

The sign minus means that resultant displacement has direction $4.63^{\circ} S\ of\ W$.

Therefore, the total displacement $d$ has magnitude 5.02 km and direction of $4.63^{\circ} S\ of\ W$.