Problem 26 - Calculus III

Question

The acceleration function, initial velocity, and initial position of a particle are \(\mathbf{a}(t)=-5 \cos t \mathbf{i}-5 \sin t \mathbf{j}, \mathbf{v}(0)=9 \mathbf{i}+2 \mathbf{j}\), and \(\mathbf{r}(0)=5 \mathbf{i}\) Find \(\mathbf{v}(t) \operatorname{and} \mathbf{r}(t)\)

Step-by-Step Solution

Answer

\( \mathbf{v}(t) = (-5 \sin t + 9) \mathbf{i} + (5 \cos t - 3) \mathbf{j} \); \( \mathbf{r}(t) = (5 \cos t + 9t) \mathbf{i} + (5 \sin t - 3t) \mathbf{j} \).

Step 1: Integrate the acceleration function

Given the acceleration function \( \mathbf{a}(t) = -5 \cos t \mathbf{i} - 5 \sin t \mathbf{j} \), we need to integrate it to find the velocity function, \( \mathbf{v}(t) \). The integration process is performed component-wise:Integrating the \( \mathbf{i} \) component:\[ \int -5 \cos t \, dt = -5 \sin t + C_1 \]Integrating the \( \mathbf{j} \) component:\[ \int -5 \sin t \, dt = 5 \cos t + C_2 \]Thus, the velocity function becomes:\[ \mathbf{v}(t) = (-5 \sin t + C_1) \mathbf{i} + (5 \cos t + C_2) \mathbf{j} \]

Step 2: Apply Initial Conditions for Velocity

We use the initial velocity \( \mathbf{v}(0) = 9 \mathbf{i} + 2 \mathbf{j} \) to find the constants \( C_1 \) and \( C_2 \).Substitute \( t = 0 \) in the velocity equation:\[ \begin{align*}\mathbf{v}(0) &= (-5 \sin 0 + C_1) \mathbf{i} + (5 \cos 0 + C_2) \mathbf{j} \9 \mathbf{i} + 2 \mathbf{j} &= (0 + C_1) \mathbf{i} + (5 + C_2) \mathbf{j} \\text{This results in:} \C_1 &= 9 \C_2 &= 2 - 5 = -3\end{align*} \]Therefore, the velocity function is:\[ \mathbf{v}(t) = (-5 \sin t + 9) \mathbf{i} + (5 \cos t - 3) \mathbf{j} \]

Step 3: Integrate the velocity function

Use the velocity function \( \mathbf{v}(t) = (-5 \sin t + 9) \mathbf{i} + (5 \cos t - 3) \mathbf{j} \) to find the position function, \( \mathbf{r}(t) \).Integrate the \( \mathbf{i} \) component:\[ \int (-5 \sin t + 9) \, dt = 5 \cos t + 9t + C_3 \]Integrate the \( \mathbf{j} \) component:\[ \int (5 \cos t - 3) \, dt = 5 \sin t - 3t + C_4 \]Thus, the position function is:\[ \mathbf{r}(t) = (5 \cos t + 9t + C_3) \mathbf{i} + (5 \sin t - 3t + C_4) \mathbf{j} \]

Step 4: Apply Initial Conditions for Position

We use the initial position \( \mathbf{r}(0) = 5 \mathbf{i} \) to determine the constants \( C_3 \) and \( C_4 \).Substitute \( t = 0 \) in the position equation:\[ \begin{align*}\mathbf{r}(0) &= (5 \cos 0 + 0 + C_3) \mathbf{i} + (5 \sin 0 - 0 + C_4) \mathbf{j} \5 \mathbf{i} &= (5 + C_3) \mathbf{i} + (0 + C_4) \mathbf{j} \\text{This gives us:} \C_3 &= 0 \C_4 &= 0\end{align*} \]Thus, the position function becomes:\[ \mathbf{r}(t) = (5 \cos t + 9t) \mathbf{i} + (5 \sin t - 3t) \mathbf{j} \]

Key Concepts

Acceleration Function

The acceleration function describes how the velocity of a particle changes over time. It is a vector function, which means it has both magnitude and direction. In the provided problem, the acceleration function is given by \( \mathbf{a}(t) = -5 \cos t \mathbf{i} - 5 \sin t \mathbf{j} \). This indicates that the particle's acceleration depends on trigonometric functions, which are periodic.
This function tells us how rapidly the velocity is changing in the \( \mathbf{i} \) (horizontal) and \( \mathbf{j} \) (vertical) components. The negative signs in front suggest the direction of acceleration is opposite to the direction of the respective trigonometric functions. Understanding the acceleration function is crucial because it serves as the starting point for finding velocity and position: integrate this function to move towards these solutions.

Velocity Function

The velocity function tells us the rate of change of the position of the particle. It's derived from the acceleration function by integration. In vector calculus, the integration of each component of the acceleration vector gives us the respective components of the velocity vector.
After integrating the provided acceleration function, we get:

• For the \( \mathbf{i} \) component: \( \int -5 \cos t \, dt = -5 \sin t + C_1 \)
• For the \( \mathbf{j} \) component: \( \int -5 \sin t \, dt = 5 \cos t + C_2 \)

Leading to the result, \( \mathbf{v}(t) = (-5 \sin t + 9) \mathbf{i} + (5 \cos t - 3) \mathbf{j} \) after applying the initial conditions. The constants \( C_1 \) and \( C_2 \) are determined using the given initial velocity, \( \mathbf{v}(0) = 9 \mathbf{i} + 2 \mathbf{j} \).

Position Function

The position function provides the location of the particle at any given time and is found by integrating the velocity function. This step involves another round of integration. The velocity function components are integrated to get the position function components:

• Integrating the \( \mathbf{i} \) component: \( \int (-5 \sin t + 9) \, dt = 5 \cos t + 9t + C_3 \)
• Integrating the \( \mathbf{j} \) component: \( \int (5 \cos t - 3) \, dt = 5 \sin t - 3t + C_4 \)

This results in the position function: \( \mathbf{r}(t) = (5 \cos t + 9t) \mathbf{i} + (5 \sin t - 3t) \mathbf{j} \). Initial conditions, \( \mathbf{r}(0) = 5 \mathbf{i} \), are used to find the constants \( C_3 \) and \( C_4 \), both equal to zero.
Understanding the position function is key, as it maps out the path of the particle over time.

Integration

Integration is a fundamental process in calculus used to find functions given their derivatives. In this problem, integration is carried out twice: once on the acceleration to find velocity, and again on velocity to find position.
Each component of the vector function is integrated separately:

• The \( \mathbf{i} \) component involves integration of \( -5 \cos t \) and later \( -5 \sin t + 9 \).
• The \( \mathbf{j} \) component involves integration of \( -5 \sin t \) and later \( 5 \cos t - 3 \).

Each integration results in an indefinite integral that includes a constant. The constants are determined using initial conditions, which provide known values at specific times, often \( t = 0 \). This process links the abstract concept of integration to the practical determination of function properties.

Initial Conditions

Initial conditions are specific values for functions or their derivatives at a particular point. They are essential for finding the constants in the solutions of integration. In this exercise, initial conditions are given for both velocity and position.
The initial velocity \( \mathbf{v}(0) = 9 \mathbf{i} + 2 \mathbf{j} \) helped determine the constants \( C_1 = 9 \) and \( C_2 = -3 \) in the velocity function. Similarly, the initial position \( \mathbf{r}(0) = 5 \mathbf{i} \) established \( C_3 = 0 \) and \( C_4 = 0 \) in the position function.
These initial conditions ensure that the velocity and position functions precisely represent the particle's state at \( t = 0 \), allowing us to accurately map its behavior over time.