Question

Suppose you first walk 12.0 m in a direction 20 owest of north and then 20.0 m in a direction 40.0osouth of west. How far are you from your starting point, and what 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 , then this problem asks you to find their sum R = A + B . The two displacements A and B add to give a total displacement R having magnitude R and direction in degrees.

Answer 1

Answer:

R=19.5m

$\theta$ = 4.65° S of W

Explanation:

Refer the attached fig.

displacement  of the x and y components

x-component displacement is ($R_{x}$) = $A_{x}+B_{x}$

= A $\sin$(20°) + B $\cos$(40°)

= -12.0$\sin$(20°) + 20.0$\cos$(40°)

= -19.425m

x-component displacement is ($R_{y}$) = $A_{y}+ B_{y}$

=  A $\cos$(20°) - B $\sin$(40°)

= 12.0$\cos$(20°) - 20.0$\sin$(40°)

= -1.579

resultant displacement

∴

R = $\sqrt{R_{x}^{2} +R_{y}^{2} } }$

=$\sqrt{(-19.425)^{2}+(-1.579)^{2} }$

=19.5m

$\theta$ = $\tan^{-1}\left | \frac{R_{x}}{R_{y}} \right |$

$\theta$ = $\tan^{-1}\left | \frac{1.579}{19.425} \right |$

$\theta$ = 4.65° S of W