Problem 83 - Calculus 3

Question

A particle travels along the path of a helix with the equation \(\mathbf{r}(t)=\cos (t) \mathbf{i}+\sin (t) \mathbf{j}+t \mathbf{k} .\) See the graph presented here: Find the following: Velocity of the particle at any time

Step-by-Step Solution

Answer

The velocity is \( \mathbf{v}(t) = -\sin(t) \mathbf{i} + \cos(t) \mathbf{j} + 1 \mathbf{k} \).

Step 1: Understand the Helix Equation

The helix is represented by the vector function \( \mathbf{r}(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k} \). This function defines the path of the particle in three dimensions as a function of time \( t \).

Step 2: Find the Derivative of the Position Function

To find the velocity of the particle, we need to differentiate the position vector function \( \mathbf{r}(t) \) with respect to time \( t \).

Step 3: Differentiate Each Component

Differentiate each component of \( \mathbf{r}(t) \): - The derivative of \( \cos(t) \) is \( -\sin(t) \).- The derivative of \( \sin(t) \) is \( \cos(t) \).- The derivative of \( t \) is \( 1 \).

Step 4: Write the Velocity Vector

Combine the derivatives to form the velocity vector \( \mathbf{v}(t) = -\sin(t) \mathbf{i} + \cos(t) \mathbf{j} + 1 \mathbf{k} \). This represents the velocity of the particle at any time \( t \).

Key Concepts

Helix Equation

The helix equation is a fascinating representation of a 3D curve, described through a vector function. In this specific example, the equation \( \mathbf{r}(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k} \) illustrates a spiral-like path, integrating circular and linear components.

What makes a helix special is its combined rotational and translational motion along the z-axis. The x and y components, \( \cos(t) \) and \( \sin(t) \), trace out a circle in the xy-plane, while the z component, \( t \), implies a simultaneous elevation along the z-axis.

This dual nature of rotation and translation gives the helix its characteristic spiral shape, as the particle moves forward while rotating around the axis. Helixes often appear in natural and engineered systems, like DNA strands or springs, making them a critical concept in fields such as biology, physics, and engineering.

Velocity Vector

The velocity vector is pivotal in understanding the motion of particles along curves like helixes. Essentially, it represents the rate of change of the particle's position with respect to time.

To obtain this velocity vector from a position vector, one must differentiate the position function. Here, for the helix described by \( \mathbf{r}(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k} \), differentiation yields the velocity vector \( \mathbf{v}(t) = -\sin(t) \mathbf{i} + \cos(t) \mathbf{j} + 1 \mathbf{k} \).

The velocity vector informs us how fast and in which direction the particle is moving at any given moment. This dynamic insight is essential for applications in physics and engineering where understanding forces and motions is crucial. The velocity vector not only shows the direction of motion but hints at how the velocity changes over time.

Differentiation

Differentiation is a core process in vector calculus used to find the rate of change of a function. In the context of motion along a helix, differentiating the position function gives us the velocity vector, offering insights into the particle's motion.

The process involves taking the derivative of each component of the position vector.

• The derivative of \( \cos(t) \) results in \( -\sin(t) \).
• The derivative of \( \sin(t) \) gives \( \cos(t) \).
• The derivative of \( t \) is \( 1 \).

These derivatives are combined to form the velocity vector.

Understanding differentiation is crucial as it is used across various domains including physics for finding velocity, acceleration, and more. It simplifies the complexity in predicting how systems evolve over time. Differentiation, as shown here, allows us to translate a position into a meaningful velocity profile, bridging the gap between static equations and dynamic motion.