91Ó°ÊÓ

The coordinate system is a fundamental tool in physics to describe the position of an object in space. It's like a map that gives coordinates to different points. In this exercise, we use a two-dimensional coordinate system, represented by the x-axis (horizontal) and y-axis (vertical). This system helps us pinpoint the car's movement as it travels from the center of town.

• The starting point, labeled as the origin, is at coordinates (0,0).
• The car's initial travel to the east is along the positive x-axis, reaching the point (80,0).
• After turning south, it moves along the y-axis to the point (80,-192).

Visualizing this movement on a graph clarifies the car's path, allowing us to see it in two individual legs: one to the east and one to the south. Recognizing this coordinate system makes it easier to define vectors and calculate displacement.

The Pythagorean theorem is a mathematical principle often used to determine the distances between two points in a coordinate system. In this context, it helps us calculate the car's total displacement, considered the shortest path back to the original point.

The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. In formula terms:
\[ c^2 = a^2 + b^2 \]

Where:

• \(c\) is the hypotenuse (displacement in this problem).
• \(a\) and \(b\) are the east and south distances traveled by the car, 80 km and 192 km respectively.

Applying this to our exercise, the car's displacement can be computed by\[ c = \sqrt{80^2 + 192^2} \], which gives us the shortest distance the car is from the starting point.

In physics, the magnitude of a vector measures how long or how far something is from the origin without considering its direction. Consider it as the length of a straight line drawn between two points in this scenario.

To find the magnitude of the displacement vector in our exercise, we use the formula derived from the Pythagorean theorem:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, the coordinates are:

• Point 1 at the origin (0,0)
• Point 2 at the car's final position (80,-192)

Substituting these into the formula gives us:\[ d = \sqrt{80^2 + (-192)^2} = \sqrt{6400 + 36864} \].

Ultimately, this calculation results in the magnitude of the displacement vector as 208 km, signifying the direct distance from the town’s center to the car’s final position.