Calculating Displacement in Vector Addition: A Practical Example

In this article, we will explore how to calculate a person's displacement when moving in a series of directions, using a practical example. We will walk you through the process, drawing a diagram and applying the Pythagorean Theorem to find the displacement or net distance traveled. This is a fundamental concept in vector addition and useful in various real-world applications, such as navigation and physics.

The Problem

A boy runs 40 meters towards the east. He then walks 30 meters towards the north. Finally, he runs 50 meters towards the north, making an angle of 30 degrees with the east. We need to find his displacement or net distance traveled.

Step-by-Step Solution

Let's break down each part of the journey and represent it graphically. We will draw a set of connected line segments to represent each part of the journey.

Step 1: The boy runs 40 meters towards the east.

Step 2: He then walks 30 meters towards the north.

Step 3: Finally, he runs 50 meters towards the north, making an angle of 30 degrees with the east.

To find the displacement, we need to determine the final position relative to the starting point. We will use the Pythagorean Theorem to find the displacement vector.

Drawing the Diagram

We will draw a diagram with a series of line segments: each representing a portion of the journey. The first line segment will be 40 meters long, pointing east. The second line segment will be 30 meters long, pointing north. The third line segment will be 50 meters long, making a 30-degree angle with the east.

Using the Pythagorean Theorem

From the diagram, we can see that the first two movements (40 meters east and 30 meters north) form a right triangle. The third movement does not align perfectly with the first two, but we can still use vector addition to find the overall displacement.

Right Triangle Analysis

The first two movements form a right triangle with the east-west axis. We can find the horizontal and vertical components of the displacement:

Horizontal Component: 40 meters east (50 meters * cos(30°))

Vertical Component: 30 meters north (50 meters * sin(30°))

Using trigonometric values:

cos(30°) √3/2

sin(30°) 1/2

Horizontal Component: 40 50 * (√3/2) 40 50 * (√3/2) 40 43.3 83.3 meters

Vertical Component: 30 50 * (1/2) 30 25 55 meters

Calculating the Displacement

Now that we have the horizontal and vertical components, we can use the Pythagorean Theorem to find the magnitude of the displacement vector:

Displacement (d) √((Horizontal Component)2 (Vertical Component)2)

Displacement (d) √((83.3)2 (55)2) √(6938.89 3025) √9963.89 ≈ 99.82 meters

Next, we need to find the direction of the displacement vector. We can use the arctangent function to find the angle:

Direction (θ) arctan(V/ H)

Direction (θ) arctan(55 / 83.3) ≈ 33.435 degrees North of East

Conclusion

The boy's displacement is 99.82 meters, 33.435 degrees North of East, or approximately 99.82 meters on a bearing of 56.565 degrees East of North.

By following this method, you can solve similar problems on your own. It's a practical way to apply vector addition in real-world scenarios.

Related Keywords

Vector Addition: Adding vectors to find the resultant displacement or force.

Displacement Calculation: The process of finding the net distance traveled based on a series of movements.

Right Triangle: A triangle with one 90-degree angle, used to apply the Pythagorean Theorem and trigonometric functions.