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In fluid particle motion, the position vector is fundamental for understanding the exact location of a particle in space at a given time. A position vector is typically represented in three dimensions: x, y, and z, which are symbolized as \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) unit vectors. In the given exercise, the position vector denotes the fluid particle moving in a spiral path, \[ \mathbf{r}_{p}(t) = 2 \sin(3t) \mathbf{i} + 2 \cos(3t) \mathbf{j} - 5t \mathbf{k}. \]

• \(2 \sin(3t) \mathbf{i}\): Represents oscillation along the x-axis.
• \(2 \cos(3t) \mathbf{j}\): Denotes oscillation along the y-axis.
• \(-5t \mathbf{k}\): Implies a consistent, linear descent along the z-axis.

When \(t = 1\) second, you substitute into the position vector to find where the fluid particle is. This gives you precise coordinates in the spiral path they are describing. Hence, by plugging \(t = 1\) into the equation, we find the exact position of the particle at that instant.

The velocity vector helps us understand how fast and in what direction the fluid particle is moving at any given moment. It is the derivative of the position vector with respect to time. This represents the rate of change of position.For the exercise, \[ \mathbf{V}(t) = \frac{d}{dt} \mathbf{r}_{p}(t) = 6 \cos(3t) \mathbf{i} - 6 \sin(3t) \mathbf{j} - 5 \mathbf{k}. \]

• \(6 \cos(3t) \mathbf{i}\): Illustrates how the x-component changes over time.
• \(-6 \sin(3t) \mathbf{j}\): Shows the changing rate along the y-axis.
• \(-5 \mathbf{k}\): Indicates a steady negative velocity, consistent along the z-axis.

For example, to find the velocity vector at \(t = 1\), substitute \(t = 1\) into the derived equation. This calculation will tell you the speed and direction of the fluid particle at that precise moment.

Acceleration vectors bring insight into how the velocity of a fluid particle changes over time. It is calculated as the derivative of the velocity vector concerning time. This provides a more in-depth understanding of the particle's motion dynamics by quantifying the change in speed and direction.Using the information in the exercise, \[ \mathbf{a}(t) = \frac{d}{dt} \mathbf{V}(t) = -18 \sin(3t) \mathbf{i} - 18 \cos(3t) \mathbf{j}. \]

• \(-18 \sin(3t) \mathbf{i}\): Describes how the x-component of the velocity changes.
• \(-18 \cos(3t) \mathbf{j}\): Represents the evolution in the y-component of velocity.

Notice that the z-component derivative is zero, indicating that the velocity component in the z-direction does not change, confirming a steady progression downward. At \(t = 1\), replacing \(t\) with 1 will provide the specific acceleration at that time.

Differential calculus is essential in understanding motion through derivatives, illustrating how functions change over time. It allows us to calculate both velocity and acceleration vectors directly from the position vector. The process involves differentiating the function that describes the position of the particle to obtain velocity, and then differentiating the velocity to obtain acceleration.Here's how it works step-by-step:1. From the position vector \( \mathbf{r}_{p}(t) \), apply the derivative to obtain \( \mathbf{V}(t) \), the velocity vector. 2. Differentiate \( \mathbf{V}(t) \) to acquire \( \mathbf{a}(t) \), the acceleration vector.This step-by-step approach displays the underlying principles of calculus:

• Illustrates the rate of change from one vector to another.
• Guides through understanding how position derivatives result in velocity and how further derivation reveals acceleration.

This application showcases the practical uses of differential calculus in describing detailed particle motion through vectors.

Spiral path motion characterizes a trajectory where a fluid particle traces a spiral similar to that in a tornado. This unique path involves simultaneous circular motion and vertical movement, depicted by the position vector provided in the exercise. The spiral path arises from the combination of sinusoidal functions for x and y, and a linear function for z: - Spiral motion in the horizontal plane is attributed to the oscillatory nature of the sine and cosine terms, creating circular patterns. - The linear term in the z-component implies vertical motion, adding a third dimension to the trajectory, creating a spiral shape. Determining the exact motion of particles along such paths helps in comprehending natural phenomena, like tornadoes, by modeling the complexities of their fluid movement. The fluid particle follows a cyclonic route reinforced through the stated position components, establishing a spiraling direction through three-dimensional space.