Velocity Component Calculation At T=1 Along A Given Vector Direction

In physics, understanding the motion of objects often requires breaking down velocity into components along different directions. This allows us to analyze the motion more effectively and predict the object's behavior. In this article, we will delve into the process of finding the component of velocity at a specific time ($t=1$) in a given direction, represented by the vector $\underline{i} - 3\underline{j} + 2\underline{k}$. This involves concepts from vector algebra and calculus, providing a comprehensive understanding of how to tackle such problems.

Understanding Velocity Components

To truly grasp the concept, velocity components are essential for dissecting motion in multiple dimensions. Velocity, a vector quantity, embodies both the speed and direction of an object's movement. In a three-dimensional space, velocity can be resolved into components along the coordinate axes (x, y, and z), typically denoted by the unit vectors $\underline{i}$, $\underline{j}$, and $\underline{k}$, respectively. These components represent the contribution of the velocity along each axis.

When we express velocity as a vector, for instance, $\underline{v} = v_x \underline{i} + v_y \underline{j} + v_z \underline{k}$, the terms $v_x$, $v_y$, and $v_z$ represent the magnitudes of the velocity components along the x, y, and z axes. These components are scalars, indicating the extent to which the velocity is directed along each axis. Breaking down velocity into components simplifies the analysis of motion, especially when dealing with projectile motion or motion on inclined planes. By considering each component separately, we can apply kinematic equations independently along each axis, making the problem more manageable. Furthermore, understanding velocity components is crucial for calculating quantities like kinetic energy and momentum in specific directions, providing a deeper insight into the dynamics of the system. The process of resolving velocity into components also plays a vital role in various fields, including robotics, computer graphics, and aerospace engineering, where precise control and prediction of motion are paramount. The ability to analyze and manipulate velocity components enables us to design and control complex systems with greater accuracy and efficiency, highlighting the fundamental importance of this concept in both theoretical and applied physics.

Determining Velocity at t=1

To begin, we need to determine the velocity vector of the object at the specified time, $t=1$. The velocity vector at any time $t$ is typically given by the derivative of the position vector $\underline{r}(t)$ with respect to time, denoted as $\underline{v}(t) = \frac{d\underline{r}(t)}{dt}$. The position vector, $\underline{r}(t)$, describes the object's location in space as a function of time. It is usually expressed in terms of its components along the coordinate axes, such as $\underline{r}(t) = x(t)\underline{i} + y(t)\underline{j} + z(t)\underline{k}$, where $x(t)$, $y(t)$, and $z(t)$ are the time-dependent coordinates of the object.

To find the velocity vector, we differentiate each component of the position vector with respect to time. This means $\underline{v}(t) = \frac{dx(t)}{dt}\underline{i} + \frac{dy(t)}{dt}\underline{j} + \frac{dz(t)}{dt}\underline{k}$. Each term in this expression represents the velocity component along the corresponding axis: $\frac{dx(t)}{dt}$ is the velocity along the x-axis, $\frac{dy(t)}{dt}$ is the velocity along the y-axis, and $\frac{dz(t)}{dt}$ is the velocity along the z-axis. Once we have found the general expression for the velocity vector $\underline{v}(t)$, we can evaluate it at the specific time $t=1$ to find the velocity at that instant. This involves substituting $t=1$ into the expressions for the velocity components. The resulting vector, $\underline{v}(1)$, gives us the instantaneous velocity of the object at $t=1$, which is crucial for analyzing its motion and predicting its future trajectory. The process of finding the velocity vector from the position vector highlights the fundamental relationship between position and velocity in kinematics, emphasizing the role of calculus in describing motion. This understanding is essential for solving a wide range of physics problems, from simple projectile motion to more complex scenarios involving variable forces and accelerations.

Projecting Velocity onto a Given Direction

The task at hand is to find the component of the velocity vector $\underline{v}(1)$ in the direction specified by the vector $\underline{d} = \underline{i} - 3\underline{j} + 2\underline{k}$. This involves the concept of vector projection, which is a way of determining how much of one vector lies along the direction of another. The component of a vector $\underline{A}$ along a vector $\underline{B}$ is given by the projection formula:

$\text{comp}_{\underline{B}} \underline{A} = \frac{\underline{A} \cdot \underline{B}}{|\underline{B}|}$

Here, $\underline{A} \cdot \underline{B}$ represents the dot product of the vectors $\underline{A}$ and $\underline{B}$, and $|\underline{B}|$ denotes the magnitude of vector $\underline{B}$. The dot product is a scalar quantity that measures the alignment between the two vectors, while the magnitude represents the length of the vector. In our case, $\underline{A}$ is the velocity vector $\underline{v}(1)$, and $\underline{B}$ is the direction vector $\underline{d} = \underline{i} - 3\underline{j} + 2\underline{k}$. The projection formula calculates the scalar component of $\underline{v}(1)$ along the direction of $\underline{d}$. This scalar value represents the magnitude of the velocity in the specified direction. To find the vector component, we multiply this scalar by the unit vector in the direction of $\underline{d}$. The unit vector is obtained by dividing $\underline{d}$ by its magnitude, resulting in a vector with a length of 1 pointing in the same direction as $\underline{d}$. The process of vector projection is fundamental in many areas of physics, including mechanics, electromagnetism, and optics. It allows us to decompose vectors into components along specific directions, simplifying the analysis of complex systems and providing a deeper understanding of the underlying physical phenomena. Understanding vector projection is also crucial in various applications, such as computer graphics, robotics, and engineering, where the decomposition of forces, velocities, and other vector quantities is essential for problem-solving and design.

Calculating the Dot Product

First, let's assume we have calculated the velocity vector at $t=1$ and found it to be $\underline{v}(1) = 2\underline{i} + \underline{j} - 3\underline{k}$. The next step is to compute the dot product of $\underline{v}(1)$ and $\underline{d}$. 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 defined as:

$\underline{A} \cdot \underline{B} = A_xB_x + A_yB_y + A_zB_z$

Applying this to our vectors $\underline{v}(1)$ and $\underline{d}$, we have:

$\underline{v}(1) \cdot \underline{d} = (2)(1) + (1)(-3) + (-3)(2) = 2 - 3 - 6 = -7$

The dot product, a scalar quantity, provides a measure of how much two vectors align with each other. It is a fundamental operation in vector algebra and has significant applications in physics and engineering. In this specific context, the dot product helps us determine the component of the velocity vector in the direction of the given vector $\underline{d}$. A positive dot product indicates that the vectors have a component in the same direction, while a negative dot product suggests they have components in opposite directions. A zero dot product implies that the vectors are orthogonal, meaning they are perpendicular to each other. The dot product is not only used to find the component of one vector along another but also to calculate the work done by a force, the angle between two vectors, and to check for orthogonality. Its properties, such as commutativity ($\underline{A} \cdot \underline{B} = \underline{B} \cdot \underline{A}$) and distributivity ($\underline{A} \cdot (\underline{B} + \underline{C}) = \underline{A} \cdot \underline{B} + \underline{A} \cdot \underline{C}$), make it a versatile tool in vector manipulations. Understanding and being able to compute the dot product is essential for anyone working with vector quantities, as it forms the basis for many calculations and analyses in various fields.

Determining the Magnitude of the Direction Vector

Next, we need to calculate the magnitude of the direction vector $\underline{d} = \underline{i} - 3\underline{j} + 2\underline{k}$. The magnitude of a vector $\underline{B} = B_x\underline{i} + B_y\underline{j} + B_z\underline{k}$ is given by:

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

For our direction vector $\underline{d}$, this becomes:

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

The magnitude of a vector is a scalar quantity that represents the length or size of the vector. It is always a non-negative value and provides a measure of the vector's intensity or strength. In the context of the direction vector $\underline{d}$, the magnitude $|\underline{d}|$ gives us the length of the vector that specifies the direction we are interested in. Calculating the magnitude is a fundamental step in many vector operations and is essential for normalizing vectors, finding unit vectors, and computing projections. The magnitude is derived from the Pythagorean theorem and extends the concept of length from two-dimensional space to three-dimensional space (or higher dimensions). It is a crucial concept in physics, engineering, and computer graphics, where vectors are used to represent various physical quantities and geometric entities. The magnitude is used in conjunction with the direction of a vector to fully characterize the vector quantity. For instance, in physics, the magnitude of a velocity vector represents the speed of an object, while the direction indicates the object's direction of motion. Similarly, the magnitude of a force vector represents the strength of the force, and the direction indicates the force's line of action. Understanding how to calculate and interpret the magnitude of a vector is essential for solving problems involving vector quantities and for developing a solid foundation in vector algebra.

Calculating the Velocity Component

Now we can compute the component of the velocity vector $\underline{v}(1)$ in the direction of $\underline{d}$ using the formula:

$\text{comp}_{\underline{d}} \underline{v}(1) = \frac{\underline{v}(1) \cdot \underline{d}}{|\underline{d}|} = \frac{-7}{\sqrt{14}}$

To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{14}$:

$\frac{-7}{\sqrt{14}} \cdot \frac{\sqrt{14}}{\sqrt{14}} = \frac{-7\sqrt{14}}{14} = -\frac{\sqrt{14}}{2}$

Thus, the component of the velocity at $t=1$ in the direction of $\underline{i} - 3\underline{j} + 2\underline{k}$ is $-\frac{\sqrt{14}}{2}$. This result represents the scalar projection of the velocity vector onto the direction vector. It indicates the magnitude of the velocity component acting along the specified direction. The negative sign suggests that this component is in the opposite direction to the vector $\underline{d}$. In physical terms, this means that the object's motion has a component that is partially against the direction defined by $\underline{d}$. This calculation is crucial in analyzing the motion of objects in three-dimensional space, as it allows us to break down the velocity into components along specific directions, making it easier to understand and predict the object's behavior. The scalar projection provides a single number that quantifies how much of the velocity vector lies along the direction of interest. This is particularly useful in scenarios where we need to determine the effect of a force or velocity component in a particular direction, such as when calculating work done or analyzing constrained motion. The ability to compute and interpret scalar projections is a fundamental skill in physics and engineering, enabling a more detailed and nuanced understanding of vector quantities and their interactions.

Conclusion

In conclusion, determining the component of velocity in a specific direction is a fundamental problem in physics that combines concepts from vector algebra and calculus. By following the steps outlined above—finding the velocity vector, calculating the dot product, and normalizing the direction vector—we can accurately determine the component of velocity in any given direction. This skill is crucial for analyzing motion in three dimensions and has wide-ranging applications in various fields of science and engineering.