Problem 166 - Calculus Volume 3

Question

A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector \(\mathbf{r}(t)=(3 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+t^{2} \mathbf{k}\). The path is similar to that of a helix, although it is not a helix. The graph is shown here: Find the following quantities: The velocity and acceleration vectors

Step-by-Step Solution

Answer

Velocity: \((-3 \sin t) \mathbf{i} + (3 \cos t) \mathbf{j} + 2t \mathbf{k}\), Acceleration: \((-3 \cos t) \mathbf{i} + (-3 \sin t) \mathbf{j} + 2 \mathbf{k}\)."

Step 1: Understand the Position Vector

The position vector is given as \( \mathbf{r}(t) = (3 \cos t) \mathbf{i} + (3 \sin t) \mathbf{j} + t^2 \mathbf{k} \). This vector describes the position of the hang glider at time \(t\). The path is a helical type, formatted as a function of time \(t\) for each component in the \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) directions.

Step 2: Find the Velocity Vector

The velocity vector \(\mathbf{v}(t)\) is the derivative of the position vector \(\mathbf{r}(t)\) with respect to time \(t\). \[ \mathbf{v}(t) = \frac{d}{dt}[ (3 \cos t) \mathbf{i} + (3 \sin t) \mathbf{j} + t^2 \mathbf{k} ] \].Calculating the derivatives for each component:- \(\frac{d}{dt}[ 3 \cos t ] = -3 \sin t\)- \(\frac{d}{dt}[ 3 \sin t ] = 3 \cos t\)- \(\frac{d}{dt}[ t^2 ] = 2t\).Thus, the velocity vector is:\[ \mathbf{v}(t) = (-3 \sin t) \mathbf{i} + (3 \cos t) \mathbf{j} + 2t \mathbf{k} \].

Step 3: Find the Acceleration Vector

The acceleration vector \(\mathbf{a}(t)\) is the derivative of the velocity vector \(\mathbf{v}(t)\) with respect to time \(t\). \[ \mathbf{a}(t) = \frac{d}{dt}[ (-3 \sin t) \mathbf{i} + (3 \cos t) \mathbf{j} + 2t \mathbf{k} ] \].Calculating the derivatives for each component:- \(\frac{d}{dt}[ -3 \sin t ] = -3 \cos t\)- \(\frac{d}{dt}[ 3 \cos t ] = -3 \sin t\)- \(\frac{d}{dt}[ 2t ] = 2\).Thus, the acceleration vector is:\[ \mathbf{a}(t) = (-3 \cos t) \mathbf{i} + (-3 \sin t) \mathbf{j} + 2 \mathbf{k} \].

Key Concepts

Understanding the Velocity Vector

The velocity vector is a crucial element in understanding motion in calculus. It tells us the speed and direction of movement for an object, which in this case is a hang glider.

To find the velocity vector, we take the derivative of the position vector with respect to time, which represents how the position changes as time progresses. The position vector of the hang glider is given by:\[ \mathbf{r}(t) = (3 \cos t) \mathbf{i} + (3 \sin t) \mathbf{j} + t^2 \mathbf{k} \]

• Derivative of the first component: The derivative of \(3 \cos t\) with respect to \(t\) is \(-3 \sin t\).
• Derivative of the second component: The derivative of \(3 \sin t\) is \(3 \cos t\).
• Derivative of the third component: The derivative of \(t^2\) is \(2t\).

Thus, the velocity vector \(\mathbf{v}(t)\) is found to be:\[ \mathbf{v}(t) = (-3 \sin t) \mathbf{i} + (3 \cos t) \mathbf{j} + 2t \mathbf{k} \]This vector indicates that the hang glider moves in a spiral path with a velocity that varies over time. Each component of \(\mathbf{v}(t)\) tells how fast and in which direction the glider moves along the \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) axes.

Exploring the Acceleration Vector

Acceleration vectors describe how quickly the velocity of an object changes. When we calculate the acceleration, we focus on the direction and magnitude of these changes.

Just as we found the velocity by taking the derivative of the position vector, the acceleration vector is the derivative of the velocity vector with respect to time. The velocity vector \(\mathbf{v}(t)\) for our hang glider is:\[ \mathbf{v}(t) = (-3 \sin t) \mathbf{i} + (3 \cos t) \mathbf{j} + 2t \mathbf{k} \]

• Derivative of the first component: The derivative of \(-3 \sin t\) is \(-3 \cos t\).
• Derivative of the second component: The derivative of \(3 \cos t\) is \(-3 \sin t\).
• Derivative of the third component: The derivative of \(2t\) is \(2\).

Thus, the acceleration vector \(\mathbf{a}(t)\) is:\[ \mathbf{a}(t) = (-3 \cos t) \mathbf{i} + (-3 \sin t) \mathbf{j} + 2 \mathbf{k} \]This means the hang glider's velocity changes differently along each axis. It shows how quickly and in which direction the speed of the glider is changing.

The Role of Derivatives in Motion

In calculus, derivatives play a fundamental role in analyzing motion through vectors. They provide a mathematical way to understand how various quantities change over time.

Derivatives help us compute key components of motion like velocity and acceleration vectors from position functions. Here are some key points:

• Velocity Vector: This derivative of the position vector gives the rate of change of location, representing speed and direction.
• Acceleration Vector: Taking the derivative of velocity, this vector describes how speed and direction change over time.

By determining derivatives, we break down complex movement into understandable chunks, showing how each aspect evolves. Thus, the velocity and acceleration vectors provide detailed insights into the motion of objects, such as a hang glider, by revealing the nature of their path and speed changes. Calculus, therefore, becomes an invaluable tool for delving deep into the dynamics of motion.