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A person walks in the following pattern: 3.3 km north, then 1.7 km west, and finally 4.3 km south.

(a) How far would a bird fly in a straight line from the same starting point to the same final point?

(b) At what angle (measured counterclockwise from east) would the bird fly?

Answer :

(a) The bird would fly approximately 1.97 km in a straight line.

(b) The bird would fly at an angle of approximately 210.5° counterclockwise from east.

  • Given:

North-South direction: The person walks 3.3 km north and then 4.3 km south.

Net displacement in North-South direction: 3.3 km - 4.3 km = -1.0 km. (negative sign indicates a direction towards the south)

East-West direction: The person walks 1.7 km west.

Net displacement in East-West direction: -1.7 km. (negative sign indicates a direction towards the west)

  • a) The magnitude of the resultant displacement can be found using the Pythagorean theorem:

[tex]D = \sqrt{(-1.0\,\text{km})^2 + (-1.7 \text{km})^2}[/tex]

[tex]D = \sqrt{1.0 + 2.89}[/tex]

[tex]D = \sqrt{3.89}[/tex]

[tex]D \approx 1.97 \text{ km}[/tex]

  • b) We use trigonometry to find the angle with respect to the positive x-axis (East direction). First, we find the angle using the tangent function:

[tex]\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{|-1.0 \text{ km}|}{|-1.7 \text{ km}|}[/tex]

[tex]\theta = \tan^{-1}\left(\frac{1.0}{1.7}\right)[/tex]

[tex]\theta \approx \tan^{-1}(0.5882)[/tex]

[tex]\theta \approx 30.5^\circ[/tex]

  • Since the direction is south of west, the actual direction measured counterclockwise from east is:

[tex]\text{Angle from East} = 180^\circ + 30.5^\circ[/tex]

[tex]= 210.5^\circ[/tex]

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