A wildlife researcher is tracking a flock of geese. The geese fly 5.0km due west, then turn toward the north by 35∘ and fly another 3.0km .
How far west are they of their initial position?
What is the magnitude of their displacement?
To find out how far west the geese are from their initial position, we need to use basic trigonometry. Here's how you can calculate it:
1. Draw a diagram: Draw a line to represent the initial position, and then another line 5.0km to the west. Now, draw a line at a 35∘ angle to the north from the end of the second line.
2. Use trigonometry: The line from step 1, which is the distance traveled west, is the adjacent side of a right triangle. The hypotenuse is the total distance traveled by the geese (5.0km). To find the westward distance, use the cosine function:
cos(35∘) = adjacent ÷ hypotenuse
Rearranging the formula to solve for the adjacent side (westward distance), we get:
adjacent = hypotenuse × cos(35∘)
Plugging in the values, we get:
adjacent = 5.0km × cos(35∘)
Calculate the value of cos(35∘) using a calculator, and multiply it by 5.0 km to find the westward distance.
To calculate the magnitude of the displacement (the straight-line distance from the initial to the final position), use the Pythagorean theorem:
displacement = √(westward distance)^2 + (3.0 km)^2
Substitute the westward distance you found earlier and calculate the value to find the magnitude of the displacement.