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1 month ago

A hiker walks 200m west and then walks 100m north. What is the magnitude and direction of her resulting displacement?

Physics

2 answers:

Ostrovityanka [3.2K]1 month ago

7 0

The calculated magnitude of the hiker's displacement is $\fbox{\begin\\224\,{\text{m}}\end{minispace}}$ , and the direction of this displacement is $\fbox{north of west direction}$ .

Further explanation:

Displacement refers to the minimal distance covered by an object from its starting point to its end location.

Given:

The distance traveled by the hiker towards the west is $200\,{\text{m}}$ .

The distance covered by the hiker towards the north is $100\,{\text{m}}$ .

Concept used:

The hiker begins at point A, moves westward to point B, then travels 100 m north to reach point C. To find the shortest path, we determine the displacement which is from point C back to point A. The path taken by the hiker can be visualized as a right triangle, labeled ABC, with AB as the base, BC as the height, and CA as the hypotenuse.

In accordance with the Pythagorean theorem, “the square of the hypotenuse is equal to the sum of the squares of the other two sides.”

The Pythagorean theorem can be expressed as:

${\left( {AC} \right)^2} = {\left( {BC} \right)^2} + {\left( {AB} \right)^2}$

To find the hypotenuse, rearrange the formula:

$\left( {AC} \right) = \sqrt {{{\left( {BC} \right)}^2} + {{\left( {AB} \right)}^2}}$ …… (1)

The angle between side AB and AC can be defined as:

$\theta={\sin ^{-1}}\left( {\dfrac{{BC}}{{AC}}} \right)$ …… (2)

Substituting $200\,{\text{m}}$ for AB and $100\,{\text{m}}$ for BC in equation (1) gives:

$\begin{aligned}\left( {AC}\right)&=\sqrt {{{\left( {200\,{\text{m}}} \right)}^2} + {{\left( {100\,{\text{m}}} \right)}^2}}\\&=223.6\,{\text{m}}\\&\simeq {\text{224}}\,{\text{m}} \\ \end{aligned}$

Similarly, substitute $100\,{\text{m}}$ for BC and $224\,{\text{m}}$ for AC in equation (2):

$\begin{aligned}\theta&={\sin ^{-1}}\left( {\frac{{100\,{\text{m}}}}{{224\,{\text{m}}}}}\right) \\&={\left( {26.51} \right)^ \circ }\\ \end{aligned}$

The final result shows that the hiker's shortest distance is $224\,{\text{m}}$ in the $\fbox{North West direction}$ at an angle of ${26.51^ \circ }$ .

Learn more:

1. Conservation of momentum
2. Motion under friction
3. Motion under gravitation

Answer Details:

Grade: High School

Subject: Physics

Chapter: Speed and distance

Keywords:

Speed, distance, displacement, velocity, time, magnitude, direction, north, east, west, south, Pythagorean theorem, triangle, sine, cosine, tangent, 224 m, 223.6 m, 26.51 deg, 27 deg.

Maru [3.3K]1 month ago

7 0

A hiker proceeds 200 m west and subsequently another 100 m north, resulting in a displacement of 223 m. The direction can be determined using the trigonometric function where sin(angle) = opposite/hypotenuse, yielding an angle of 26.6 degrees. Therefore, the total displacement is 223 m at an angle of 26.6 degrees north of west.

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