Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider the motion of a particle along a helix given by \(\mathbf{r}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+\left(t^{2}-3 t+2\right) \mathbf{k}\), where the \(\mathbf{k}\) component measures the height in meters above the ground and \(t \geq 0\). If the particle leaves the helix and moves along the line tangent to the helix when it is 12 meters above the ground, give the direction vector for the line.

Step-by-Step Solution

Answer

Direction vector is \( \cos(5) \mathbf{i} - \sin(5) \mathbf{j} + 7 \mathbf{k}. \)

Step 1: Find when the particle is 12 meters above the ground

We need to solve for when the \[ t^2 - 3t + 2 = 12. \]This simplifies to:\[ t^2 - 3t + 2 - 12 = 0, \]which further simplifies to:\[ t^2 - 3t - 10 = 0. \]

Step 2: Solve the quadratic equation

To solve the quadratic equation \( t^2 - 3t - 10 = 0 \), use the quadratic formula:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]where \( a = 1 \), \( b = -3 \), and \( c = -10 \). Then\[ t = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1}. \]This results in\[ t = \frac{3 \pm \sqrt{9 + 40}}{2}, \]\[ t = \frac{3 \pm \sqrt{49}}{2}. \]Therefore, \( \sqrt{49} = 7 \), and\[ t = \frac{3 + 7}{2} \quad \text{or} \quad t = \frac{3 - 7}{2}. \]Thus, \( t = 5 \) or \( t = -2 \). Since \( t \geq 0 \), \( t = 5 \).

Step 3: Determine the tangent vector at \( t = 5 \)

Find the derivative of \( \mathbf{r}(t) = \sin t \mathbf{i} + \cos t \mathbf{j} + (t^2 - 3t + 2) \mathbf{k} \) with respect to \( t \):\[ \mathbf{r}'(t) = \cos t \mathbf{i} - \sin t \mathbf{j} + (2t - 3) \mathbf{k}. \]Substitute \( t = 5 \) to find the tangent vector:\[ \mathbf{r}'(5) = \cos(5) \mathbf{i} - \sin(5) \mathbf{j} + (2\times5 - 3) \mathbf{k}. \]

Step 4: Calculate the components of the tangent vector

Calculate the components at \( t = 5 \):- \( \mathbf{i} \) component: \( \cos(5) \)- \( \mathbf{j} \) component: \( -\sin(5) \)- \( \mathbf{k} \) component: \( 2 \cdot 5 - 3 = 7 \)Thus, the tangent vector at \( t = 5 \) is:\[ \cos(5) \mathbf{i} - \sin(5) \mathbf{j} + 7 \mathbf{k}. \]

Step 5: Finalize the direction vector

The direction vector for the line tangent to the helix when the particle is 12 meters above the ground is \( \cos(5) \mathbf{i} - \sin(5) \mathbf{j} + 7 \mathbf{k} \).

Key Concepts

Parametric Equations

Parametric equations are fundamental in describing motion, especially in cases that don't conform to single-variable functions. In this context, we have the motion of a particle along a helical path, dictated by the parametric equation \( \mathbf{r}(t) = \sin t \mathbf{i} + \cos t \mathbf{j} + (t^2 - 3t + 2) \mathbf{k} \). This equation involves three parameters that individually describe movement in the horizontal plane (\( \mathbf{i} \) and \( \mathbf{j} \) directions) and vertical movement (\( \mathbf{k} \) direction). By using parametric equations, we can track the position of the particle in all three spatial dimensions simultaneously.
Parametric equations provide several advantages:

• They allow for the representation of complex 3D paths that would be difficult to analyze using standard Cartesian equations.
• They offer flexibility to independently vary each component over time, which is crucial for accurately modeling motion.

These equations are instrumental in physics and engineering, helping in visualizing and solving problems involving trajectories, such as the problem at hand.

Tangent Vectors

Tangent vectors play a vital role in understanding the instantaneous direction of motion of a particle traveling along a path, such as a helix. In our exercise, we derive the tangent vector by differentiating the parametric equation \( \mathbf{r}(t) \) with respect to time \( t \).
To find the tangent vector, we compute the derivative:

• \( \mathbf{r}'(t) = \cos t \mathbf{i} - \sin t \mathbf{j} + (2t - 3) \mathbf{k} \)

This vector represents the direction in which the particle is moving at a given point in time. Calculating this at \( t = 5 \) involves substituting \( t \) to get the specific tangent vector when the particle reaches 12 meters in height.
Understanding tangent vectors is important as they:

• Help in determining the path's direction at a specific point.
• Allow for finding equations for lines tangent to curves, critical in geometry and physics.

Essentially, they enable us to make predictions about the trajectory of moving particles.

Quadratic Equations

Quadratic equations frequently appear in problems involving motion, like those in kinematic studies, where variables change with time. Here, a quadratic equation helps solve for the time \( t \) when the height of the particle is 12 meters. The equation
\( t^2 - 3t + 2 = 12 \)
Is transformed into a standard quadratic form:

• \( t^2 - 3t - 10 = 0 \)

We apply the quadratic formula:

• \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

With values \( a = 1 \), \( b = -3 \), and \( c = -10 \), yielding a solution for \( t \):

• \( t = 5 \)

for the physical scenario of \( t \geq 0 \).
Quadratic equations are cornerstone elements in mathematical analysis:

• They emerge naturally in problems involving acceleration and forces.
• Provide solutions through straightforward and logical calculations, as demonstrated in solving for time \( t \).

They are indispensable tools for solving and interpreting real-world phenomena, including particle motion.

Particle Motion

Motion of particles, like our particle traveling along a helix, is vital in physics and engineering. Understanding particle motion involves analyzing how position changes over time, using models like the parametric equations described.
In our problem, we examine the trajectory of the particle, especially focusing on when it leaves the helical path to follow a tangent line. This requires understanding both the spatial position and the velocity direction at the departure point. Key aspects include:

• Identifying when a particle reaches a specific height (12 meters in this case), utilizing components of motion equations.
• Calculating and understanding the tangent vector, which gives insight into the immediate directionality of motion

Studying particle motion enables us to make forecasts about where and how fast particles move. It's a basis for advanced applications, including:

• Analyzing trajectories in mechanics and aerodynamics.
• Designing paths for moving components in machinery and robotics.

In essence, this exercise provides a practical introduction to the principles governing particle motion.