91Ó°ÊÓ

The Pythagorean theorem is a fundamental concept in math and physics, used to determine the length of the sides of a right triangle. Imagine you have a triangle where two sides meet at a right angle. These are your "a" and "b" sides.
The hypotenuse, "c," is the side opposite the right angle, and it's the longest side of the triangle. The theorem states:

• \[ c^2 = a^2 + b^2 \]

This means that the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides.
For example, if you know the lengths of two sides, you can always find the length of the third side by rearranging the equation. It's perfect for finding distances like in our displacement problem, where we need to find the hypotenuse, i.e., the net displacement.

Vectors are used to describe quantities like displacement that have both direction and magnitude. A vector can be broken down into components, which are the projections of that vector onto the coordinate axes. Typically, we consider horizontal (x-axis) and vertical (y-axis) components.
In our problem, the student's movements east and west contribute to the x-component, while the northward movement affects the y-component.

• The total movement east is 30 m (positive x-direction).
• The total movement west is 50 m (negative x-direction).
• Finally, movement north is 40 m (positive y-direction).

By calculating the resultant x and y components, we can simplify complex movements into straightforward calculations. These components are essential for finding the net displacement using vector addition and the Pythagorean theorem.

Net displacement refers to the shortest distance from the starting point to the final position of an object. It is determined by calculating the vector sum of all individual displacements.
For our scenario:

• We find the resultant x-component, which is \(-20\) m, as the student moved 30 m east and then 50 m west.
• The y-component is straightforward—40 m north, as no southward movement negates it.

With these components, we employ the Pythagorean theorem to find the net displacement. This involves treating the resultant x and y components as the two shorter sides of a right triangle and solving for the hypotenuse:

• Calculate: \( d = \sqrt{(-20)^2 + (40)^2} \).
• This equates to: \( d = \sqrt{400 + 1600} = \sqrt{2000} \), resulting in \( d = 44.72 \text{ m} \).

Thus, the correspondence to options provided in the exercise, this displacement lies between 40 m and 60 m, confirming the solution.