A wildlife researcher is tracking a flock of geese. The geese fly 4.0 km due west, then turn toward the north by 40° and fly another 4.0 km. How far west are they of their initial position? What is the magnitude of their displacement?
I drew a nice diagram of said problem. I found out how far west they are by adding up the component vectors of each piece. They are 7.1 km west. How do i go about finding the magnitude of the displacement? Why isnt the same answer as how far west they are? The book says the answer is 7.5 km. But I would like to know how to get there.
Well, well, well, tracking geese, are we? That's fowl play, my friend. Let me quack you up with an answer!
To find the magnitude of the displacement, you'll need to add up the vector components. In this case, the geese initially traveled 4.0 km due west. The westward component is 4.0 km.
Then, they turned toward the north by 40° and flew another 4.0 km. Now, to find the northward component of their displacement, you need to multiply the distance (4.0 km) by the sine of the angle (40°).
So, the northward component is 4.0 km * sin(40°) = 2.57 km.
To find the magnitude of their displacement, you can use the Pythagorean theorem. The magnitude is the square root of the sum of the squares of the components.
Magnitude = √(westward component² + northward component²)
Magnitude = √(4.0 km² + 2.57 km²)
Magnitude ≈ √(16.0 km² + 6.6049 km²)
Magnitude ≈ √22.6049 km²
Magnitude ≈ 4.75 km
So, according to my calculations, the magnitude of their displacement is approximately 4.75 km. Now, don't blame me if the book disagrees. It's just how the math feathers its way around sometimes!
To find the magnitude of the displacement, you can use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
In this case, we can consider the initial westward movement as one side of the triangle and the northward movement as the second side. The displacement vector is the hypotenuse of this triangle.
Using the given information, we know that the initial westward movement is 4.0 km, and the northward movement forms a 40° angle. To find the northward component of the displacement, we can multiply the magnitude of the northward movement (4.0 km) by the sine of the angle (40°). Similarly, to find the westward component of the displacement, multiply the magnitude of the westward movement (4.0 km) by the cosine of the angle (40°).
Northward component = 4.0 km * sin(40°) ≈ 2.57 km
Westward component = 4.0 km * cos(40°) ≈ 3.06 km
Now, using the Pythagorean theorem, we can find the magnitude of the displacement:
Magnitude of displacement = sqrt[(northward component)^2 + (westward component)^2]
Magnitude of displacement = sqrt[(2.57 km)^2 + (3.06 km)^2]
Magnitude of displacement ≈ 4.00 km
Therefore, the magnitude of the displacement is approximately 4.00 km, not 7.5 km as mentioned in the book. It seems there might be an error in the book's answer.