In this article, we will explore the concept of resultant displacement in a straightforward and detailed manner. We will analyze a scenario where a boy walks in multiple directions and determine the total displacement. Understanding these principles is crucial for anyone interested in physics and navigation. Let's dive in.

Understanding Displacement

Displacement is a vector quantity that refers to the change in position of an object. It has both magnitude and direction. In this article, we will focus on finding the resultant displacement of a boy walking in different directions.

Problem Statement

The problem presented is as follows: A boy walks 5 meters east, then 4 meters south, and finally 2 meters west. We need to determine the magnitude and direction of the resultant displacement.

Step-by-Step Solution

Step 1: Break Down the Displacements

Let's break down the individual displacements into their respective components:

5 meters east (5m East) 4 meters south (4m South) 2 meters west (2m West)

Step 2: Calculate Net East-West Displacement

The net displacement in the east-west direction is:

5 meters East - 2 meters West  3 meters East

Step 3: Calculate Net North-South Displacement

The net displacement in the north-south direction is straightforward since there is no northward movement:

4 meters South

Step 4: Determine the Magnitude of the Resultant Displacement

The magnitude of the resultant displacement is calculated using the Pythagorean theorem:

Resultant Displacement  √(3^2   4^2)  √(9   16)  √25  5 meters

Step 5: Determine the Direction of the Resultant Displacement

To find the direction, we use the tangent function to determine the angle:

tan(θ)  opposite / adjacent  4 / 3θ  arctan(4/3)  arctan(1.333) ≈ 53° 7′ 48.4″

This angle is measured from the east direction, with south being positive, giving us a direction of 53° 7′ 48.4″ South of East.

Visualization

The path of the boy can be sketched as follows (visual representation not shown, but the green vector represents the resultant displacement)

Conclusion

The magnitude of the resultant displacement is 5 meters, and the direction is 53° 7′ 48.4″ South of East. Understanding these concepts is crucial for fields such as navigation, physics, and mathematics.

In summary, the key steps involve breaking down the displacements, calculating the net east-west and north-south displacements, and then using the Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant displacement.