Problem 13 - Essential Calculus Early Transcendentals

Question

(a) Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. (b) Use a computer to graph the path of the particle. $$\mathbf{a}(t)=2 t \mathbf{i}+\sin t \mathbf{j}+\cos 2 t \mathbf{k}, \quad \mathbf{v}(0)=\mathbf{i}, \quad \mathbf{r}(0)=\mathbf{j}$$

Step-by-Step Solution

Answer

(a) \( \mathbf{r}(t) = \frac{t^3}{3} \mathbf{i} + (t + \sin t + 1) \mathbf{j} - \left( \frac{1}{4}(\cos 2t - 1) \right) \mathbf{k} \). (b) Use a 3D graphing tool for visualization.

Step 1: Integrate the Acceleration to find Velocity

The acceleration vector is given by \( \mathbf{a}(t) = 2t \mathbf{i} + \sin t \mathbf{j} + \cos 2t \mathbf{k} \). To find the velocity vector \( \mathbf{v}(t) \), we need to integrate the acceleration vector with respect to time. This gives:\[\mathbf{v}(t) = \int \mathbf{a}(t) \, dt = \int (2t \mathbf{i} + \sin t \mathbf{j} + \cos 2t \mathbf{k}) \, dt.\]Integrating each component separately, we get:\[\mathbf{v}(t) = \left( t^2 + C_1 \right) \mathbf{i} + \left( -\cos t + C_2 \right) \mathbf{j} + \left( \frac{1}{2} \sin 2t + C_3 \right) \mathbf{k}\]where \( C_1, C_2, \text{and} \ C_3 \) are constants of integration.

Step 2: Apply Initial Velocity to find Constants

We know from the problem that \( \mathbf{v}(0) = \mathbf{i} \). This will help us find the constants \( C_1, C_2, \text{and} \ C_3 \) in the velocity function. Substituting \( t = 0 \) into the equation for \( \mathbf{v}(t) \), we get:\[\mathbf{v}(0) = (0 + C_1) \mathbf{i} + (-\cos 0 + C_2) \mathbf{j} + \left( \frac{1}{2} \sin 0 + C_3 \right) \mathbf{k} = \mathbf{i}.\]This implies \( C_1 = 1 \), \( C_2 = 1 \), and \( C_3 = 0 \). So, the velocity vector simplifies to:\[\mathbf{v}(t) = (t^2 + 1) \mathbf{i} + (-\cos t + 1) \mathbf{j} + \frac{1}{2} \sin 2t \mathbf{k}\]

Step 3: Integrate Velocity to find Position

Now we integrate the velocity vector \( \mathbf{v}(t) = (t^2 + 1) \mathbf{i} + (-\cos t + 1) \mathbf{j} + \frac{1}{2} \sin 2t \mathbf{k} \) to find the position vector \( \mathbf{r}(t) \):\[\mathbf{r}(t) = \int \mathbf{v}(t) \, dt = \int ((t^2 + 1) \mathbf{i} + (-\cos t + 1) \mathbf{j} + \frac{1}{2} \sin 2t \mathbf{k}) \, dt.\]Integrating each component separately, we find:\[\mathbf{r}(t) = \left( \frac{t^3}{3} + t + D_1 \right) \mathbf{i} + (\sin t + t + D_2) \mathbf{j} + \left( -\frac{1}{4} \cos 2t + D_3 \right) \mathbf{k}\]where \( D_1, D_2, \text{and} \ D_3 \) are constants of integration.

Step 4: Apply Initial Position to find Constants

We use \( \mathbf{r}(0) = \mathbf{j} \) to determine the new constants:\[\mathbf{r}(0) = \left( \frac{0^3}{3} + 0 + D_1 \right) \mathbf{i} + (\sin 0 + 0 + D_2) \mathbf{j} + \left( -\frac{1}{4} \cos 0 + D_3 \right) \mathbf{k} = \mathbf{j}.\]This gives us \( D_1 = 0 \), \( D_2 = 1 \), and \( D_3 = \frac{1}{4} \). The position vector is:\[\mathbf{r}(t) = \frac{t^3}{3} \mathbf{i} + (t + \sin t + 1) \mathbf{j} - \left( \frac{1}{4}(\cos 2t - 1) \right) \mathbf{k}\]

Step 5: Visualize the Path using a Computer

To visualize the path, we plot the trajectory of the particle over a range of time values using a computer. This involves plotting the vector function \( \mathbf{r}(t) \) in 3D space. Software like MATLAB, Python with Matplotlib, or any graphing tool that supports 3D plotting can be used to achieve this visualization.

Key Concepts

Acceleration Vectors

Acceleration vectors play a crucial role in understanding how a particle's speed and direction change over time. In this exercise, the acceleration vector is given by \( \mathbf{a}(t) = 2t \mathbf{i} + \sin t \mathbf{j} + \cos 2t \mathbf{k} \). This vector indicates that the acceleration changes with time across three dimensions, represented by the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). Each component of the acceleration vector determines how the particle accelerates along each axis.

To find the velocity of the particle, we need to integrate the acceleration vector. This involves finding the antiderivative for each component. This process transforms information about how the object's velocity is changing into information about the velocity itself. Integration is essential in this context as it allows us to compute motion properties when acceleration is known.

It's important to understand that the constants obtained during integration (constants of integration) are determined using initial conditions, which are provided in the problem as the initial velocity of the particle. Understanding acceleration vectors is a building block in dealing with more complex motion scenarios.

Velocity Vectors

Velocity vectors represent the speed and direction of a particle at any given time. As derived in the exercise, the velocity vector \( \mathbf{v}(t) \) is obtained by integrating the acceleration vector, leading to \( \mathbf{v}(t) = (t^2 + 1) \mathbf{i} + (-\cos t + 1) \mathbf{j} + \frac{1}{2} \sin 2t \mathbf{k} \). This vector tells us how the position of the particle changes over time along the \( x \), \( y \), and \( z \) axes.

Each component of the velocity vector contains terms resulting from the integration of the respective acceleration components. These terms can include polynomials, trigonometric functions, and constants. The constant terms reflect initial conditions like the particle's starting velocity. The process ensures that the calculated velocity matches the particle's behavior when \( t = 0 \).

• The \( \mathbf{i} \) component governs how the velocity changes along the x-axis.
• The \( \mathbf{j} \) component controls the change along the y-axis.
• The \( \mathbf{k} \) component manages the variation along the z-axis.

By understanding velocity vectors, we grasp not only the speed of particle motion but also the directional shifts occurring over time.

Integration

Integration is a fundamental concept in calculus used to determine one function when given its derivative. In this particle motion exercise, integration helps calculate both velocity and position vectors from given acceleration vectors.

The process involves finding antiderivatives for each component function of the acceleration vector. This isn't just a mathematical operation but a translation of what we can measure directly (acceleration) into quantities that describe more comprehensive motion aspects (velocity and position).

• By integrating the acceleration vector \( \mathbf{a}(t) \), we find the velocity vector \( \mathbf{v}(t) \).
• A further integration of the velocity vector gives us the position vector \( \mathbf{r}(t) \).

The constants of integration are determined by initial conditions, as they provide specific values that tailor the general solution to the problem at hand. Without such constants, these solutions could fit an infinite number of scenarios, highlighting the importance of initial conditions in motion studies.

Particle Motion

Particle motion refers to how a particle travels through space over time. In the context of this exercise, the particle's path is determined by its position vector \( \mathbf{r}(t) \), derived by integrating the velocity vector. The position vector is \( \mathbf{r}(t) = \frac{t^3}{3} \mathbf{i} + (t + \sin t + 1) \mathbf{j} - \frac{1}{4}(\cos 2t - 1) \mathbf{k} \). This expression tells how the particle's position evolves in three-dimensional space over time.

Understanding particle motion requires a synthesis of acceleration, velocity, and position vectors. These vectors work together to describe the complete trajectory of the particle.

• The position vector accumulates effects from both velocity and acceleration, reporting the particle's exact location.
• Each integration step provides deeper insights into the physical journey the particle undertakes.
• Visualizing this trajectory using graphing tools can provide a clearer understanding of the path, as it translates mathematical expressions into a physical representation.

This comprehensive understanding of particle motion allows us to predict future positions and reconstruct past movements based on present data.