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?
Starting at (0,0), their final location is
-4 + 4cis140° = -7.06 + 2.57i
see what you can do with that
come back if you get stuck
To determine how far west the geese are from their initial position, we need to calculate the horizontal component of their displacement.
First, let's break down the geese's movement into two parts:
1. Flying 4.0 km due west.
2. Turning towards the north by 40° and flying another 4.0 km.
To find the horizontal component of the displacement, we can use trigonometry. The horizontal displacement is equal to the distance they flew due west minus the distance they flew north.
1. To calculate the distance they flew west, we can use the cosine function. The formula for finding the horizontal component of a vector is given by:
horizontal displacement = distance * cosine(angle)
In this case, the distance is 4.0 km, and the angle is 0° since they flew directly west.
So, the horizontal displacement = 4.0 km * cosine(0°).
By substituting the values into the formula, we get:
horizontal displacement = 4.0 km * cos(0°) = 4.0 km * 1 = 4.0 km.
Therefore, the geese are 4.0 km west of their initial position.
2. Now, let's calculate the distance they flew north. To find the vertical component of the displacement, we can use the sine function. The formula is:
vertical displacement = distance * sine(angle)
In this case, the distance is also 4.0 km, and the angle is 40°.
So, the vertical displacement = 4.0 km * sine(40°).
By substituting the values into the formula, we get:
vertical displacement = 4.0 km * sin(40°) = 4.0 km * 0.64279 = 2.57 km (rounded to two decimal places).
Therefore, the vertical displacement is approximately 2.57 km.
To find the magnitude of their displacement, we can use the Pythagorean theorem:
magnitude = sqrt(horizontal displacement^2 + vertical displacement^2)
= sqrt((4.0 km)^2 + (2.57 km)^2)
= sqrt(16.0 km^2 + 6.6049 km^2)
= sqrt(22.6049 km^2)
= 4.75 km (rounded to two decimal places).
Therefore, the geese are 4.0 km west of their initial position, and the magnitude of their displacement is approximately 4.75 km.
To find out how far west the geese are from their initial position, we can use trigonometry. Let's break down the steps:
Step 1: Flying due west
The geese fly 4.0 km due west. Since they are flying directly west, there is no northward or southward component to their displacement. Therefore, their westward displacement is 4.0 km.
Step 2: Changing direction
After flying due west, the geese turn toward the north by 40°. This means that the remaining distance they fly will have a northward component.
Step 3: Finding the northward component
To find the northward component of the displacement, we can use trigonometry. Since the geese have turned 40° north from west, we can use the sine function.
The northward component = (Total displacement) x sin(angle)
The total displacement of the remaining distance is also 4.0 km since they fly another 4.0 km. So,
Northward component = 4.0 km x sin(40°)
Calculating,
Northward component ≈ 4.0 km x 0.6428 ≈ 2.5712 km
Step 4: Finding the magnitude of displacement
To find the magnitude of the displacement, we can use the Pythagorean theorem. The magnitude of displacement is the square root of the sum of the squares of the two components (westward and northward).
Magnitude of displacement = sqrt[(westward component)^2 + (northward component)^2]
Replacing the values,
Magnitude of displacement = sqrt[(4.0 km)^2 + (2.5712 km)^2]
Calculating,
Magnitude of displacement ≈ sqrt[16 km^2 + 6.618 km^2] ≈ sqrt[22.618 km^2] ≈ 4.758 km
Therefore, the geese are 4.0 km west of their initial position, and the magnitude of their displacement is approximately 4.758 km.