A cross-country skier skis 1.00km north and then 2.00km east on a horizontal snow field. How far and in what direction is she from the starting point?
The cross-country skier is approximately 2.236 kilometres from her starting point, and she is located in a direction that is approximately 26.57 degrees north of east. This information helps us precisely describe her position and direction relative to the starting point on the snow field.
First, let’s outline the given information or extract our data from the question:
- The skier starts by skiing 1.00 kilometres north.
- Then, she skis 2.00 kilometres east.
We need to find two key pieces of information: the distance from her starting point and the direction in which she is located.
To solve this problem, we can use vector addition to determine the net displacement of the skier. Vector addition is a mathematical technique that allows us to combine vectors (in this case, the two legs of her journey) to find the resultant vector, which represents her overall displacement.
Step 1: Break Down the Displacements into Components
First, we need to break down the two displacements (1.00 km north and 2.00 km east) into their respective north and east components. We’ll use the convention that north is the positive y-direction, and east is the positive x-direction.
The northward displacement is +1.00 km in the y-direction, and the eastward displacement is +2.00 km in the x-direction.
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Step 2: Apply Vector Addition
Now, we will apply vector addition to find the net displacement (R), which is the hypotenuse of a right triangle formed by the north and east components.
We can use the Pythagorean theorem, which states that in a right triangle:
R² = (Δx)² + (Δy)²
- R is the net displacement.
- Δx is the change in the x-direction (east).
- Δy is the change in the y-direction (north).
In this case:
Δx = 2.00 km (east)
Δy = 1.00 km (north)
Plugging these values into the Pythagorean theorem:
R² = (2.00 km)² + (1.00 km)²
We will now have: R² = 4.00 km² + 1.00 km²
R² = 5.00 km²
Taking the square root of both sides:
R = √5.00 km
R ≈ 2.236 km
So, the skier is approximately 2.236 kilometres from her starting point.
Step 3: Find the Direction
Now, let’s find the direction of the net displacement. We can use trigonometry to determine the angle θ that the net displacement vector makes with the positive x-axis (east).
We can use the tangent function, which is defined as:
tan(θ) = (Δy) / (Δx)
Plugging in the values:
tan(θ) = (1.00 km) / (2.00 km)
Now, take the arctan (inverse tangent) of both sides to solve for θ:
θ = arctan(1.00/2.00)
Using a calculator:
θ ≈ 26.57 degrees
So, the direction of the net displacement is approximately 26.57 degrees north of east.
She will be located in a direction that is approximately 26.57 degrees north of east.