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Zorluk: OrtaDirection and Distance Test

An archaeological surveyor is mapping an excavation site using a miscalibrated compass. The compass needle that is supposed to point 'North' actually points to the true geographical East.

Starting from the base camp, she faces the direction the compass indicates as 'North' and walks 20 m20\text{ m} straight. She then turns 135∘135^\circ clockwise and walks 102 m10\sqrt{2}\text{ m} in a straight line. Next, she turns 45∘45^\circ anti-clockwise and proceeds 10 m10\text{ m} forward. Finally, she turns 90∘90^\circ clockwise and walks another 10 m10\text{ m} to reach the main artifact chamber.

What is the true geographical direction and shortest direct distance of the main artifact chamber relative to the base camp?

  1. 20 m20\text{ m}, SouthCevap
  2. B
    20 m20\text{ m}, East
  3. C
    20 m20\text{ m}, North
  4. D
    105 m10\sqrt{5}\text{ m}, South-East

Cevap

The main artifact chamber is 20 m20\text{ m} directly to the South of the base camp.
By applying each movement as a vector on a coordinate plane, the starting path of 20 m20\text{ m} East is followed by 102 m10\sqrt{2}\text{ m} South-West (yielding −10-10 East, −10-10 North). The surveyor then goes 10 m10\text{ m} South, and finally 10 m10\text{ m} West. Summing the horizontal (East/West) components: +20−10+0−10=0+20 - 10 + 0 - 10 = 0. Summing the vertical (North/South) components: 0−10−10+0=−200 - 10 - 10 + 0 = -20. The net displacement is exactly 20 m20\text{ m} South.

Adım Adım Çözüm

1
Determine the true starting direction based on the faulty compass.
The surveyor walks 20 m20\text{ m} towards the true East.
The compass's 'North' actually points East. Walking 20 m20\text{ m} in that direction results in a displacement of +20 m+20\text{ m} on the East-West axis (Coordinate: 20 East, 0 North).
2
Calculate the direction and displacement of the second movement.
She turns 135∘135^\circ clockwise to face South-West, walking 102 m10\sqrt{2}\text{ m}.
Facing East, a 135∘135^\circ clockwise turn lands exactly on South-West. A movement of 102 m10\sqrt{2}\text{ m} South-West equates to moving 10 m10\text{ m} West and 10 m10\text{ m} South (using a 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle). New coordinate: (20−10)(20 - 10) East, (0−10)(0 - 10) North = 10 m10\text{ m} East, 10 m10\text{ m} South.
3
Calculate the direction and displacement of the third movement.
She turns 45∘45^\circ anti-clockwise to face South, walking 10 m10\text{ m}.
Facing South-West, turning 45∘45^\circ anti-clockwise (left) brings her to face South. Walking 10 m10\text{ m} South brings her coordinate to 10 m10\text{ m} East, 20 m20\text{ m} South.
4
Calculate the direction and displacement of the final movement.
She turns 90∘90^\circ clockwise to face West, walking 10 m10\text{ m}.
Facing South, a 90∘90^\circ clockwise turn points West. Moving 10 m10\text{ m} West shifts her East-West position by −10 m-10\text{ m}. Final coordinate: (10−10)(10 - 10) East, 20 m20\text{ m} South = 0 m0\text{ m} East, 20 m20\text{ m} South.
5
Determine the final distance and direction from the starting point.
The final position is exactly 20 m20\text{ m} South of the base camp.
The East-West displacements cancel out completely, leaving a pure Southward displacement of 20 m20\text{ m}.

Anahtar Kavram

Vector displacement, angular rotation tracing, and relative reference framing.
Tahmini Süre:1m 30s
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Konular

Direction and Distance Test

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