Calculating the Resultant Velocity of Multiple Vectors: A Comprehensive Guide

Understanding how to calculate the resultant velocity of multiple vectors is crucial in physics and engineering. This guide will walk you through the process of calculating the resultant velocity of three specific vectors: 8 m/s due south, 12 m/s due east, and 3√2 m/s due northeast. By the end of this article, you'll have a clear understanding of vector addition and the component method.

Introduction to Vector Addition

Vector addition is a fundamental concept in physics and engineering. It involves finding the resultant vector, which represents the overall effect of multiple vectors. The process involves breaking down each vector into its x and y components, adding corresponding components, and then using the Pythagorean theorem to find the magnitude of the resultant vector. The direction can be determined using the tangent function.

Breaking Down the Velocities

Let's consider the three given velocities:

8 m/s due south 12 m/s due east 3√2 m/s due northeast

To find the resultant velocity, we will break down each vector into its x and y components. This process is illustrated below:

The x-components of the Velocities

0 m/s from the 8 m/s due south 12 m/s from the 12 m/s due east 3√2 m/s × cos45° 3√2/2 m/s from the 3√2 m/s due northeast

The total x-component of the velocities is:

12 3√2/2 15√2/2 m/s

The y-components of the Velocities

-8 m/s from the 8 m/s due south 0 m/s from the 12 m/s due east 3√2 m/s × sin45° 3√2/2 m/s from the 3√2 m/s due northeast

The total y-component of the velocities is:

-8 3√2/2 -5√2/2 m/s

Calculating the Resultant Velocity

Now that we have the x and y components, we can calculate the magnitude of the resultant velocity using the Pythagorean theorem:

Resultant velocity √(15√2/2)^2 (-5√2/2)^2

Resultant velocity √(225/4 25/4)

Resultant velocity √(250/4) 5√10 m/s

Determining the Direction

To determine the direction, we use the inverse tangent function:

Direction tan^-1(-5√2/2 / 15√2/2) tan^-1(-1/3) -18.43° from the positive x-axis or 71.57° from the positive x-axis.

Therefore, the direction and magnitude of the resultant velocity are 71.57° from the positive x-axis and 5√10 m/s, respectively.

Concluding Thoughts

The process of calculating the resultant velocity is a powerful tool in physics and engineering. Understanding the component method and vector addition is essential for mastering more advanced topics in these fields. By applying the steps outlined in this guide, you can confidently tackle more complex problems involving multiple vectors.