Velocity Component Calculation At T=1 A Step-by-Step Solution

Introduction

In the realm of physics and engineering, understanding the motion of objects is paramount. This often involves analyzing velocity, which is a vector quantity describing the rate of change of an object's position. Velocity has both magnitude (speed) and direction. When analyzing motion in three dimensions, we often need to determine the component of velocity in a specific direction. This is crucial for understanding how the object is moving along a particular axis or trajectory. In this article, we will delve into the process of finding the component of velocity at a specific time in a given direction, which is a fundamental concept in kinematics and dynamics. We will explore the mathematical tools and techniques required to solve such problems, providing a comprehensive guide for students and professionals alike. Understanding velocity components is essential for applications ranging from projectile motion analysis to spacecraft navigation.

Problem Statement

The problem at hand involves finding the component of velocity at a specific time, $t = 1$, in the direction of a given vector, $\underline{i} - 3\underline{j} + 2\underline{k}$. The velocity vector is given by $4\underline{i} + 2\underline{j}$. This is a classic problem in vector algebra and calculus, often encountered in introductory physics and engineering courses. To solve this, we need to project the velocity vector onto the given direction vector. The projection will give us the component of the velocity that lies along the specified direction. This involves using the dot product of the two vectors and normalizing the direction vector. The ability to calculate velocity components is vital for predicting the motion of objects under various conditions. By breaking down the velocity into components, we can analyze motion in different directions independently, making complex problems more manageable. This problem provides a practical application of vector algebra and serves as a building block for more advanced topics in dynamics and control systems. The solution requires a clear understanding of vector operations and their geometric interpretations.

Mathematical Background

To solve this problem, we need to understand the concept of vector projection. The projection of a vector $\underline{A}$ onto a vector $\underline{B}$ is the component of $\underline{A}$ that lies in the direction of $\underline{B}$. Mathematically, the projection of $\underline{A}$ onto $\underline{B}$ is given by:

$$\text{proj}_{\underline{B}} \underline{A} = \frac{\underline{A} \cdot \underline{B}}{\|\underline{B}\|^2} \underline{B}$$

Where:

• \underline{A} \cdot \underline{B}$ is the dot product of vectors $\underline{A}$ and $\underline{B}$.

• \|\underline{B}\|$ is the magnitude of vector $\underline{B}$.

The dot product of two vectors $\underline{A} = A_x \underline{i} + A_y \underline{j} + A_z \underline{k}$ and $\underline{B} = B_x \underline{i} + B_y \underline{j} + B_z \underline{k}$ is given by:

$$\underline{A} \cdot \underline{B} = A_x B_x + A_y B_y + A_z B_z$$

The magnitude of a vector $\underline{B} = B_x \underline{i} + B_y \underline{j} + B_z \underline{k}$ is given by:

$$\|\underline{B}\| = \sqrt{B_x^2 + B_y^2 + B_z^2}$$

These formulas are the foundation for solving the problem. Mastering these concepts is crucial for understanding vector projections and their applications in physics and engineering. The dot product provides a way to quantify the alignment of two vectors, while the magnitude gives the length of a vector. By combining these tools, we can decompose a vector into its components along different directions, which is a powerful technique for analyzing motion and forces. The projection formula allows us to isolate the component of a vector that is relevant to a particular direction, simplifying the analysis and providing valuable insights.

Solution

Given the velocity vector $\underline{V} = 4\underline{i} + 2\underline{j}$ at $t = 1$ and the direction vector $\underline{D} = \underline{i} - 3\underline{j} + 2\underline{k}$, we want to find the component of $\underline{V}$ in the direction of $\underline{D}$.

First, we calculate the dot product of $\underline{V}$ and $\underline{D}$:

$$\underline{V} \cdot \underline{D} = (4)(1) + (2)(-3) + (0)(2) = 4 - 6 + 0 = -2$$

Next, we calculate the magnitude squared of $\underline{D}$:

$$\|\underline{D}\|^2 = (1)^2 + (-3)^2 + (2)^2 = 1 + 9 + 4 = 14$$

Now, we can find the projection of $\underline{V}$ onto $\underline{D}$:

$$\text{proj}_{\underline{D}} \underline{V} = \frac{\underline{V} \cdot \underline{D}}{\|\underline{D}\|^2} \underline{D} = \frac{-2}{14} (\underline{i} - 3\underline{j} + 2\underline{k}) = -\frac{1}{7} (\underline{i} - 3\underline{j} + 2\underline{k})$$

So, the component of the velocity at $t = 1$ in the direction of $\underline{i} - 3\underline{j} + 2\underline{k}$ is:

$$-\frac{1}{7} \underline{i} + \frac{3}{7} \underline{j} - \frac{2}{7} \underline{k}$$

This solution demonstrates the application of vector projection to find the component of velocity in a specific direction. The negative sign indicates that the component is in the opposite direction to the given vector. Accurate calculation of vector projections is crucial for solving problems in mechanics and other areas of physics. This step-by-step solution highlights the importance of understanding the underlying mathematical principles and applying them systematically. By breaking down the problem into smaller steps, we can avoid errors and gain a deeper understanding of the concepts involved. The final result provides valuable information about the object's motion along the specified direction.

Conclusion

In conclusion, we have successfully found the component of the velocity vector at $t = 1$ in the direction of the given vector $\underline{i} - 3\underline{j} + 2\underline{k}$. This was achieved by applying the concept of vector projection, which involves calculating the dot product of the velocity vector and the direction vector, and then normalizing the direction vector. The result, $-\frac{1}{7} \underline{i} + \frac{3}{7} \underline{j} - \frac{2}{7} \underline{k}$, represents the component of the velocity that lies along the specified direction. Understanding and applying vector projection is a fundamental skill in physics and engineering, with applications in a wide range of problems involving motion, forces, and fields. This problem serves as a valuable example of how vector algebra can be used to solve practical problems in mechanics. By mastering these techniques, students and professionals can gain a deeper understanding of the physical world and develop the skills necessary to analyze and predict the behavior of complex systems. The ability to decompose vectors into components is a powerful tool that simplifies the analysis of motion and forces, making it easier to solve problems and gain insights into the underlying physics.

Option Analysis

Based on our calculations, neither option (a) $4 \underline{i}+2 \underline{j}$ nor option (b) $4 \underline{i}-2 \underline{j}$ is the correct answer. The correct component of the velocity in the given direction is $-\frac{1}{7} \underline{i} + \frac{3}{7} \underline{j} - \frac{2}{7} \underline{k}$. This highlights the importance of performing accurate calculations and not relying solely on the given options. Careful analysis and step-by-step solutions are essential for avoiding errors and arriving at the correct answer. In this case, the options provided do not match the calculated result, indicating a potential error in the problem statement or the options themselves. This underscores the need for critical thinking and independent verification of results. By thoroughly understanding the underlying concepts and applying them correctly, we can identify discrepancies and ensure the accuracy of our solutions.