Problem 6 - Essential Calculus Early Transcendentals

Question

Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=t \mathbf{i}+2 \cos t \mathbf{j}+\sin t \mathbf{k}, \quad t=0$$

Step-by-Step Solution

Answer

Velocity is \( \mathbf{i} + \mathbf{k} \), acceleration is \(-2 \mathbf{j}\), and speed is \( \sqrt{2} \).

Step 1: Find the Velocity

To find the velocity of the particle, we need to differentiate the position vector function \( \mathbf{r}(t) \) with respect to \( t \). The position function is \( \mathbf{r}(t) = t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k} \). Thus, the velocity \( \mathbf{v}(t) \) is:\[ \mathbf{v}(t) = \frac{d}{dt}(t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k}) = \mathbf{i} - 2 \sin t \mathbf{j} + \cos t \mathbf{k}. \]

Step 2: Evaluate Velocity at Specified Time

Substitute \( t=0 \) into the velocity vector to find \( \mathbf{v}(0) \):\[ \mathbf{v}(0) = \mathbf{i} - 2 \sin(0) \mathbf{j} + \cos(0) \mathbf{k} = \mathbf{i} + \mathbf{k}. \]

Step 3: Find the Acceleration

Acceleration is the derivative of the velocity vector \( \mathbf{v}(t) \) with respect to \( t \). Differentiate the velocity function:\[ \mathbf{a}(t) = \frac{d}{dt}(\mathbf{i} - 2 \sin t \mathbf{j} + \cos t \mathbf{k}) = -2 \cos t \mathbf{j} - \sin t \mathbf{k}. \]

Step 4: Evaluate Acceleration at Specified Time

Substitute \( t=0 \) into the acceleration vector to find \( \mathbf{a}(0) \):\[ \mathbf{a}(0) = -2 \cos(0) \mathbf{j} - \sin(0) \mathbf{k} = -2 \mathbf{j}. \]

Step 5: Find the Speed

Speed is the magnitude of the velocity vector. Calculate the magnitude of \( \mathbf{v}(0) \):\[ \text{Speed} = \| \mathbf{v}(0) \| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{2}. \]

Step 6: Sketch and Interpret

First, sketch the three-dimensional path based on the position function, observing that it varies linearly in \( x \) and oscillates in \( y \) and \( z \). At \( t=0 \), the position is \( \mathbf{r}(0) = \mathbf{i} + 2 \mathbf{j} \). Draw the vectors \( \mathbf{v}(0) = \mathbf{i} + \mathbf{k} \) and \( \mathbf{a}(0) = -2 \mathbf{j} \) starting at this point. These vectors indicate the direction and rate of change of velocity and speed, respectively.

Key Concepts

Velocity of a Particle

In the context of calculus and physics, velocity in a three-dimensional space is a vector quantity that represents the rate of change of a particle's position with respect to time. It combines both the speed and direction of the particle. The position function given here is \( \mathbf{r}(t) = t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k} \).
To find the velocity vector, which is denoted by \( \mathbf{v}(t) \), you calculate the derivative of the position vector \( \mathbf{r}(t) \) with respect to time \( t \). Through differentiation, we find that: \[ \mathbf{v}(t) = \frac{d}{dt}(t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k}) = \mathbf{i} - 2 \sin t \mathbf{j} + \cos t \mathbf{k}. \]Here, each component of \( \mathbf{v}(t) \) signifies a directional change:- \( \mathbf{i} \): Change along the x-axis.- \( -2 \sin t \mathbf{j} \): Change along the y-axis.- \( \cos t \mathbf{k} \): Change along the z-axis.These components help in mapping the particle's path over time in 3D space.

Acceleration of a Particle

Acceleration is another vector quantity and represents the rate of change of velocity with respect to time. It offers insights into how a particle's velocity alters as it traverses its path. To determine the acceleration vector for the given position function, we need to find the derivative of the velocity function \( \mathbf{v}(t) \):\[ \mathbf{a}(t) = \frac{d}{dt}(\mathbf{i} - 2 \sin t \mathbf{j} + \cos t \mathbf{k}) = -2 \cos t \mathbf{j} - \sin t \mathbf{k}. \]The acceleration vector \( \mathbf{a}(t) \) can be broken down as follows:- The \(-2 \cos t \mathbf{j}\) term highlights the radial acceleration component along the y-axis, reflecting periodic slowing and speed-up as influenced by the cosine function.- The \(-\sin t \mathbf{k}\) component shows the influence along the z-axis, changing per the sine function's periodicity.These help in comprehending how the particle's velocity is not steady and changes over the three-dimensional path.

Position Vector

A position vector serves as a detailed description of a point in space. It effectively captures a particle's position in terms of its distance from an origin point in a 3D coordinate system. The given position vector for our exercise is \( \mathbf{r}(t) = t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k} \).This vector breaks down as:- \( t \mathbf{i} \) describes linear movement along the x-axis, where the particle continues onward as time progresses.- \( 2 \cos t \mathbf{j} \) adds a periodic oscillatory motion along the y-axis, owing to the trigonometric cosine function.- \( \sin t \mathbf{k} \) gives a smooth oscillation up and down along the z-axis, under the sway of the sine function.By analyzing these components, we comprehend how the particle moves via a mixture of constant motion and oscillations, crafting a unique path through 3D space.

Three-Dimensional Path

Understanding the three-dimensional path of a particle involves piecing together information about its position vector, velocity, and acceleration. The position function \( \mathbf{r}(t) = t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k} \) helps in outlining this path.Let's break this down:

• The particle moves linearly forward in the direction of the x-axis due to the term \( t \mathbf{i} \).
• It simultaneously oscillates back and forth in the y-axis caused by \( 2 \cos t \mathbf{j} \).
• Additionally, there’s an up and down motion along the z-axis influenced by \( \sin t \mathbf{k} \).

By evaluating these motions at specific points in time (like \(t=0\)), we can sketch a visual representation of the path. This insight is instrumental in illustrating the path the particle crafts through its coordinated linear and oscillatory movements. During \(t=0\), these vectors dictate positioning, velocity, and acceleration, allowing players to graphically map out the particle's trajectory in space.