Problem 87 - Calculus 3

Question

A particle travels along the path of an ellipse with the equation \(\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+0 \mathbf{k} .\) Find the following: Velocity of the particle

Step-by-Step Solution

Answer

\( \mathbf{v}(t) = -\sin t \mathbf{i} + 2 \cos t \mathbf{j} + 0 \mathbf{k} \).

Step 1: Identify the Position Function

The position function of the particle is given as \( \mathbf{r}(t) = \cos t \mathbf{i} + 2 \sin t \mathbf{j} + 0 \mathbf{k} \). This represents the position of the particle at any time \( t \).

Step 2: Determine the Velocity Function

Velocity is the derivative of the position vector with respect to time. To find the velocity \( \mathbf{v}(t) \), we need to differentiate each component of \( \mathbf{r}(t) \) with respect to \( t \).

Step 3: Differentiate the i-component

The \( \mathbf{i} \) component is \( \cos t \). The derivative of \( \cos t \) with respect to \( t \) is \( -\sin t \).

Step 4: Differentiate the j-component

The \( \mathbf{j} \) component is \( 2 \sin t \). Differentiating \( 2 \sin t \) with respect to \( t \) gives \( 2 \cos t \).

Step 5: Differentiate the k-component

The \( \mathbf{k} \) component is 0. The derivative of a constant is 0, so this component remains \( 0 \mathbf{k} \).

Step 6: Combine Derivatives to Form Velocity Vector

Combine the derivatives calculated: \( \mathbf{v}(t) = -\sin t \mathbf{i} + 2 \cos t \mathbf{j} + 0 \mathbf{k} \). This represents the velocity of the particle at any time \( t \).

Key Concepts

Position vector

In physics and mathematics, the position vector plays an important role in describing the location of a particle in space as it moves over time. For the given problem, the position vector is defined as \( \mathbf{r}(t) = \cos t \mathbf{i} + 2 \sin t \mathbf{j} + 0 \mathbf{k} \). This equation provides a clear representation of where the particle is at any given time \( t \).
The position vector has three components, each associated with a different spatial dimension:

• \( \cos t \mathbf{i} \) represents the position on the x-axis.
• \( 2 \sin t \mathbf{j} \) represents the position on the y-axis.
• \( 0 \mathbf{k} \) indicates there is no movement along the z-axis.

By analyzing these components, one can determine the trajectory of the particle as it travels through space, specifically following an elliptical path in this scenario.

Derivative of vectors

The derivative of a vector is fundamental when analyzing motion, as it leads us to understand changes in position over time, also known as velocity. To find the velocity of a particle moving along a path given by a position vector, taking the derivative with respect to time reveals much about the object's motion.
Here's how this is executed for each component in the exercise:

• The \( \mathbf{i} \) component: \( \cos t \rightarrow -\sin t \).
• The \( \mathbf{j} \) component: \( 2 \sin t \rightarrow 2 \cos t \).
• The \( \mathbf{k} \) component: \( 0 \rightarrow 0 \).

Thus, the velocity vector is \( \mathbf{v}(t) = -\sin t \mathbf{i} + 2 \cos t \mathbf{j} + 0 \mathbf{k} \).
This method helps us track how fast and in what direction the particle is moving at any given time.

Parametric equations

Parametric equations are used to express a set of related quantities as explicit functions of an independent variable, often time. In particle motion, parametric equations provide a powerful means to describe the path and movement by using time as an intermediary variable.
In the given exercise, the parametric equation \( \mathbf{r}(t) = \cos t \mathbf{i} + 2 \sin t \mathbf{j} + 0 \mathbf{k} \) helps to describe the position of the particle as it moves along the path of an ellipse:

• \( \cos t \) and \( \sin t \) are periodic functions representing oscillations over time.
• Multiplying \( \sin t \) by 2 gives it a different range or amplitude, shaping the elliptical path.

These equations allow us not only to model the path comprehensively but also to easily compute derivatives to determine velocity and acceleration.

Ellipse path

Understanding the shape of the path the particle travels along is crucial in problems involving motion. In this problem, the path forms an ellipse. An ellipse is a geometrical shape that resembles a flattened circle and can be described using parametric equations.
In our exercise, the position vector maps the particle's path as an ellipse using the equation \( \mathbf{r}(t) = \cos t \mathbf{i} + 2 \sin t \mathbf{j} + 0 \mathbf{k} \). This equation demonstrates how:

• The x-component \( \cos t \) reflects a consistent circular motion.
• The y-component \( 2 \sin t \) stretches this circular motion vertically by a factor of 2, forming an ellipse.

Such paths are prevalent in physics and engineering, particularly in systems where periodic or oscillatory behavior is analyzed, like planetary orbits or electrical circuits.