Problem 53 - Calculus III

Question

A particle travels along the path of an ellipse with the equation \(\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+0 \mathbf{k}\). Find the following:Speed

Step-by-Step Solution

Answer

The speed of the particle is \( \sqrt{1 + 3\cos^2 t} \).

Step 1: Differentiate the Position Vector

The speed of the particle is given by the magnitude of its velocity vector. To find the velocity vector \( \mathbf{v}(t) \), differentiate the position vector \( \mathbf{r}(t) = \cos t \mathbf{i} + 2 \sin t \mathbf{j} + 0 \mathbf{k} \) with respect to \( t \). The derivative is: \( \mathbf{v}(t) = \frac{d}{dt} (\cos t \mathbf{i} + 2 \sin t \mathbf{j} ) = -\sin t \mathbf{i} + 2 \cos t \mathbf{j} \).

Step 2: Find the Magnitude of the Velocity Vector

The speed of the particle is the magnitude of the velocity vector \( \mathbf{v}(t) \). Calculate this magnitude: \[ \|\mathbf{v}(t)\| = \sqrt{(-\sin t)^2 + (2 \cos t)^2} = \sqrt{\sin^2 t + 4 \cos^2 t} \].

Step 3: Simplify the Expression

Simplify the expression for the speed. Recall the identity \( \cos^2 t + \sin^2 t = 1 \):\[ \|\mathbf{v}(t)\| = \sqrt{4\cos^2 t + \sin^2 t} = \sqrt{4(1-\sin^2 t) + \sin^2 t} = \sqrt{4-3\sin^2 t}\].

Step 4: Use Trigonometric Identity to Further Simplify

Using the fundamental identity \( \sin^2 t = 1-\cos^2 t \), we can further simplify:\[ \|\mathbf{v}(t)\| = \sqrt{4 - 3(1-\cos^2 t)} = \sqrt{4-3 + 3\cos^2 t} = \sqrt{1 + 3\cos^2 t} \].

Step 5: Final Step: Conclude with the Speed Function

Conclude the process by summarizing the final expression for the speed of the particle: The speed of the particle is \( \sqrt{1 + 3\cos^2 t} \).

Key Concepts

Velocity Vector Differentiation

When studying the motion of particles along curved paths, one fundamental concept is the velocity vector differentiation. This involves taking the derivative of the position vector to obtain the velocity vector.
In the case of our particle moving along an ellipse, the position vector \( \mathbf{r}(t) = \cos t \mathbf{i} + 2 \sin t \mathbf{j} + 0 \mathbf{k} \) describes the path.
To determine the velocity, we differentiate each component with respect to time \( t \).
This calculation gives us the velocity vector \( \mathbf{v}(t) = -\sin t \mathbf{i} + 2\cos t \mathbf{j} \).
Ultimately, differentiating the position vector is about finding the rate of change of position, which is precisely what velocity describes. This step is essential in understanding motion dynamics, allowing us to further analyze aspects like speed.

Magnitude of a Velocity Vector

Finding the speed of a moving particle along a path involves calculating the magnitude of its velocity vector, which is obtained from the derivative of the position vector.
The magnitude of the velocity vector represents the actual speed of the particle without regard to direction.
In our example, where the particle follows an elliptical path, we calculated the velocity vector as \( \mathbf{v}(t) = -\sin t \mathbf{i} + 2\cos t \mathbf{j} \).
The magnitude is then determined as \[ \|\mathbf{v}(t)\| = \sqrt{(-\sin t)^2 + (2\cos t)^2} = \sqrt{\sin^2 t + 4\cos^2 t} \] This step provides the scalar speed, indicating how fast the particle travels along its path.
Understanding the magnitude of the velocity vector is crucial for linking the algebraic representation of motion with the physical sensation of speed.

Trigonometric Identities

Trigonometric identities play a pivotal role in simplifying complex mathematical expressions, especially when dealing with motion and coordinate systems.
In our example, after determining the magnitude of the velocity vector, we came across the expression \( \sqrt{\sin^2 t + 4\cos^2 t} \).
By recalling the fundamental identity \( \cos^2 t + \sin^2 t = 1 \), we simplify the expression further, turning it into \[ \|\mathbf{v}(t)\| = \sqrt{4\cos^2 t + \sin^2 t} = \sqrt{4-3\sin^2 t} \]Trigonometric identities like \( \sin^2 t = 1 - \cos^2 t \) allow the expression to be further reduced to \[ \|\mathbf{v}(t)\| = \sqrt{1 + 3\cos^2 t} \]Identities streamline these calculations, making it easier to reach a simplified and useful result that describes the motion of the particle in a clear way. This simplification helps us in formulating the final speed function, capturing the essence of the particle's motion on the ellipse.