A wildlife researcher is tracking a flock of geese. The geese fly 5.0 km due west, then turn toward the north by 45 ∘ and fly another 3.5 km .

Question

A wildlife researcher is tracking a flock of geese. The geese fly 5.0 km due west, then turn toward the north by 45 ∘ and fly another 3.5 km .

How far west are they of their initial position?

What is the magnitude of their displacement?

Answer 1

This type of question is best approached from the lens of "vector addition." I am going to describe this more or less from the ground up, so feel free to skim until you hit the point where you are encountering new information.

Vectors are objects with magnitude (size) and direction. In this problem, we are adding two length vectors:

A, a vector with length 5 km pointed West and

B, a vector with length 3.5 km pointed 45 degrees North of West.

Note: Vectors are usually bolded or written with an arrow drawn over them to signify that they are vectors.

The easiest approach to adding vectors is to first break the vectors into what are called "components." This means taking a vector and breaking it up into how far it stretches in each direction.

Breaking A into components is pretty easy:

A = -5 km [East] + 0 km [North] [note: generally we consider "East" and "North" the "positive" directions, so 5km (West) is usually expressed as -5km [East]. In this problem, calling this "+5km [West]" would not have caused any issues, but it might not be the best habit]

For the first component of A, "5 km" is our magnitude (length), and "-[East]" is its direction. For the second component, "0" is the magnitude, and "[North]" is its direction.

For B, we have to draw a right triangle. The hypotenuse is always the length of our vector, in this case 3.5 km. The hypotenuse starts at the origin and travels the same direction the vector does (really, we're just drawing the vector itself at this point). If we draw our axes such the the x-axis is pointed towards the East, and the y-axis the pointed towards the North, that means our hypotenuse will be a 3.5 km line at an angle 45 degrees above the West (AKA -East) axis.

We can now use some straightforward trig (SOH CAH TOA) to find out the two shorter sides of our triangle: The West (Adjacent) part will be 3.5 km (Hypotenuse) * cos(45 degrees), and the North (Opposite) part will be 3.5 km (Hypotenuse) * sin(45 degrees).

sin(45) = cos(45) = sqrt(2)/2 ≈ 0.707

0.707 * 3.5 km ≈ 2.47 km

Putting this together, we now get an expression for B in component form:

B = -2.47 km [East] + 2.47 [North]

A = -5 km [East] + 0 km [North] [as a reminder]

Breaking these into components was the hard part, and the reward for our hard work is that adding these together is super simple now! We simply add their common components. In fact, you might have noticed we already did this when breaking A and B up into components: We made each one out of two different vectors, one pointing in the East-West direction and another pointing in the North-South direction. I'm going to introduce another handy convention used when talking about components:

A_x = "the x component of A".

Similarly, and more relevant to our problem here, A_East = "the East component of A."

If we call define C as the vector we get when we add A + B,

C = (A_East + B_East) [East] + (A_North + B_North) [North]

Using the values from our problem,

C = (-5 km - 2.47 km) [East] + (0 km + 2.47 km) [North]

C = (-7.47 km) [East] + (2.47 km) [North

We now have what we call the "resultant" vector; the result of adding our two vectors. This tells us the total distance we moved from our starting point to our ending point, and it is in a form that makes answering the two questions being asked here nice and easy.

For how far West we traveled, we look at C_East. This equals -7.47 km, which means the geese traveled 7.47 km east in total.

For the magnitude of displacement, we just need to realize that C is going to form a triangle with C as the hypotenuse, C_East as the lower leg, and C_North as the side leg. This lets us employ the Pythagorean Theorem to find our hypotenuse:

C² = C_East² + C_North² [Note: I did not bold C here because, though one could and this would not be wrong, when we square vectors we actually lose the direction component! Think about it: (+1)² = +1, (-1)² = +1. This is okay though; we are only trying to find the length of C right now anyway]

Evaluating this for C gets us that the length of C ≈ 7.87 km. This length is the magnitude of displacement of the geese.

We don't need to worry about finding the direction in the problem, but if we did the angle between the East-West axis and C would be solvable with several trig approaches, the easiest being

tan(θ) = C_North/|C_East| [We take the absolute value of C_East to avoid ambiguity of which side is negative, dealing with issues of domains of trig functions, and since we know where our angle is going we can just force all angles to be positive values between 0 and 180 degrees and treat this as a normal "solve for an angle in a triangle" problem.]

I hope this was helpful, and if you still have any questions about how to approach this type of problem (or would like a diagram to accompany this one), please do not hesitate to reach out!