Question

A ferry leaves the terminal and travels 124km in a direction 12.0 north of west to island A. After a short stop, the ferry travels 85.4km in a direction 26.0 west of south to island B. The ferry then completes the round trip by returning directly to the terminal.
a)Draw a vector diagram that describes the entire journey.
b)What is the displacement (magnitude and direction) of the ferry in the last trip?

Answer 1

a) The diagram is presented below:

b) First, find the total displacement along the horizontal axis when the ferry traveled from zero to A:

$$d_{1x}=124\text{ cos}12^\circ.$$

From A to B along WE axis:

$$d_{2x}=84.5\text{ sin}26^\circ.$$

Now find these displacements along vertical axis:

$$d_{1y}=124\text{ sin}12^\circ.$$

$$d_{2y}=84.5\text{ cos}26^\circ.$$

Be careful, the last term is in "negative" vertical direction (down).

Now find the total displacement along horizontal and vertical:

$$d_{x}=d_{1x}+d_{2x}=\\=124\text{ cos}12^\circ+84.5\text{ sin}26^\circ=158\text{ km}.$$

$$d_{y}=d_{1y}+d_{2y}=\\=124\text{ sin}12^\circ-84.5\text{ cos}26^\circ=-50\text{ km}.$$

The total displacement during the last trip:

$$d=\sqrt{d_x^2+d_y^2}=166\text{ km}.$$

And the angle (west of south):

$$\alpha=\text{atan}\frac{d_x}{d_y}=72.0^\circ.$$