Question

suppose you first walk 12.0 m in a direction degrees west of north and then 20.0 m in a direction 40.0 degrees south of west. How far are you from your starting point, and whar is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B, as in figure 3.54, then this problem finds their sum R= A+B.)

Answer 1

The direction of 12 m is towards west north with an angle of 20 with the y-axis or north.

The direction of 20 m is towards the south west with an angle of 40 with the x-axis or west.

From the given figure,

The angle between the 12 m and the west is,

`\begin{gathered} \alpha=90^(\circ)-20^(\circ) \\ \alpha=70^(\circ) \end{gathered}`

Thus, the angle between the 12 m and 20 m is,

`\begin{gathered} \theta=\alpha+40^(\circ) \\ \theta=70^(\circ)+40^(\circ) \\ \theta=110^(\circ) \end{gathered}`

The resultant of the 12 m walk and 20 m walk is,

`R=\sqrt[]{A^2+B^2+2AB\cos (\theta)}`

Substituting the known values,

`\begin{gathered} R=\sqrt[]{12^2+20^2+2*12*20*\cos (110^(\circ))} \\ R=19.5\text{ m} \end{gathered}`

Thus, the magnitude of the resultant of R is 19.5 m.