One day you go hiking at a nearby nature preserve. At first, you follow the stra

Question

One day you go hiking at a nearby nature preserve. At first, you follow the straight, clearly marked trails. From your starting point, you travel 2.00 miles down the 1st trail. Then you turn to your left by 30.0° to follow a 2nd trail for 1.50 miles. Next, you turn to your right by 160° and follow a 3rd trail for 2.00 miles. At this point you are getting very tired and would like to get back as quickly as possible, but all of the available trails seem to lead you deeper into the woods. You would like to take a shortcut directly through the woods (ignoring the trails). How far to your right should you turn, and how far do you have to walk, to go directly back to your starting point? Feel free to use the provided vector drawing board to help visualize your work.

Explanation / Answer

Considering north and east as positive,

And taking the initial direction of the person as east,

Net displacement in the eastern direction is

2.0 + 1.5*cos 30 + 2.0*cos (360 - 160 + 30)

= 2.0 + 1.299 - 1.285

= 2.014 m toward east

Net displacement in the northern direction is

1.5*sin 30 + 2.0*sin (360 -160 + 30)

= - 0.782

Minus indicates that he has moved to south

Resultant

squae root (2.014^2 + [- 0.782^2])

= 2.160

2.160 * cos tita = 2.014

tita = 21.18 dedree south of east (below x axis)

2.160 * sin tita = - 0.782

tita = - 21.23 degree

tan tita = - 0.782/2.014

tita = - 21.22 degree from the positive x axis

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