Question

A person walks in the following pattern: \(3.1 \mathrm{~km}\) north, then \(2.4 \mathrm{~km}\) west, and finally \(5.2 \mathrm{~km}\) south. ( \(a\) ) Construct the vector diagram that represents this motion. (b) How far and in what direction would a bird fly in a straight line to arrive at the same final point?

Short answer

A bird would fly approximately 4.70 km in a direction roughly to the west of south.

Step-by-step solution

Graphic Representation

Draw a vector diagram where each of the individual walks (in the directions stated) are represented as individual vectors. This forms a triangle, where each side of the triangle represents a different leg of the walk.

Determining the Resultant Vector

Analyze the vector diagram and draw a line from the starting point (origin) directly to the final point. This straight line represents the bird's flight; it is the resultant vector. The direction of this straight line is noted for the final answer.

Calculating the Distance and Direction

By using vector algebra or the Pythagorean theorem (for right triangles), calculate the length of the resultant vector. This length represents the direct distance the bird would fly. Measure the direction of the line with respect to the North.

Key concepts

Resultant Vector

When dealing with motion in a plane, a resultant vector plays a crucial role. It is the single vector that has the same effect as multiple vectors taken together. Imagine taking a series of turns while walking: one vector represents each segment of your walk. The resultant vector is then the direct path from your starting point to your destination without all the detours.

To find this in our exercise, the steps outline drawing the vectors for each leg of the journey and then drawing a straight line from start to finish. This straight line is the resultant vector. It's the shortcut the bird in the problem would take, bypassing all the turns and walking, making it a much shorter path.

Vector Algebra

Vector algebra allows us to perform calculations on vectors similar to how we handle numbers. Vectors not only have magnitude but also direction, making them distinct from regular numbers. In vector algebra, we have operations like vector addition, subtraction, and multiplication by scalars.

In the context of our exercise, vector algebra is used to combine the multiple legs of the person's walk to find the resultant vector. Each leg of the journey is represented as a vector, and through vector algebra, we can determine the most efficient path, which is the direct distance and angle from start to end.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, specifically for right-angled triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In our problem's solution, we can use the Pythagorean theorem if we consider the north and west walks as two sides of a right triangle. Then, the bird's flight path forms the hypotenuse. The theorem then allows us to calculate the straight-line distance (hypotenuse), which is the magnitude of the resultant vector, efficiently and accurately.

Displacement

Displacement in physics is not just about how far an object has traveled; it's about the object's overall change in position. It's a vector quantity, considering both the magnitude (how far) and the direction of the movement.

When the person in our exercise walks north, then west, and finally south, they're covering a certain distance. However, their displacement would be the direct distance from where they started to where they ended, regardless of the path taken. Using vector diagrams and algebra, we can determine this displacement, which is exactly what the bird’s flight path in our problem represents.

Vector Addition

In physics, when we talk about vectors, addition becomes a way to combine these quantities. Unlike simply adding numbers, vector addition considers both magnitude and direction. The resultant of two or more vectors is found by effectively stacking them head to tail and drawing a vector from the starting point to the end.

For our walking person's example, to find the cumulative effect of walking north, then west, then south, we would add these vectors together. This combination yields a new vector which can be computed using the graphical method described in the solution steps or through algebraic methods such as using components and trigonometry.