Problem 9 - Thomas Calculus

Question

Give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. \(x=t, \quad y=\sqrt{4-t^{2}} ; \quad 0 \leq t \leq 2\)

Step-by-Step Solution

Answer

The path is the top-right quarter of the circle \(x^2 + y^2 = 4\) from \((0,2)\) to \((2,0)\).

Step 1: Identify Parametric Equations

The parametric equations given for the motion of a particle in the \(xy\)-plane are \(x = t\) and \(y = \sqrt{4 - t^2}\). The parameter \(t\) ranges from 0 to 2.

Step 2: Derive Cartesian Equation

To find the Cartesian equation, we eliminate the parameter \(t\). Since \(x = t\), we can substitute \(x\) for \(t\) in the second equation, yielding \(y = \sqrt{4 - x^2}\). Thus, the Cartesian equation is \(y = \sqrt{4 - x^2}\).

Step 3: Simplify and Identify the Particle's Path

The equation \(y = \sqrt{4 - x^2}\) represents the top half of a circle centered at the origin with a radius of 2. This is because \(x^2 + y^2 = 4\) describes a full circle, and the square root restricts \(y\) to non-negative values.

Step 4: Graph the Cartesian Equation

In this step, we plot the equation \(x^2 + y^2 = 4, y \geq 0\) on a coordinate plane. This graph is a semicircle on the top half of the circle with radius 2, where \(x\) ranges from -2 to 2. However, due to the parameter interval \(0 \leq t \leq 2\), only the segment from \((0, 2)\) to \((2, 0)\) on the semicircle is traced.

Step 5: Indicate Direction of Motion

The direction of motion for the particle follows the increase of parameter \(t\) from 0 to 2. As \(t\) increases, \(x = t\) begins at 0 and ends at 2. Thus, the particle moves along the semicircle from point \((0, 2)\) to point \((2, 0)\) in a rightward direction.

Key Concepts

Cartesian Equations

Cartesian equations relate two variables, typically referred to as \(x\) and \(y\), using only algebraic expressions. These equations describe paths or shapes in the Cartesian coordinate plane. What makes them unique is their reliance solely on the two variables without additional parameters.
In the context of the original exercise, we start with parametric equations \(x = t\) and \(y = \sqrt{4 - t^2}\). The goal is to express the relationship between \(x\) and \(y\) without involving \(t\). By substituting \(t\) as \(x\), the equation becomes \(y = \sqrt{4 - x^2}\). This simplification helps us identify the type of curve in the plane.
In this example, the transformation reveals that \(y = \sqrt{4 - x^2}\) describes the upper half of a circle centered at the origin with a radius of 2, which is a result of recognizing that \(x^2 + y^2 = 4\). This understanding makes graphing and analyzing the function straightforward.

Particle Motion

Understanding particle motion involves analyzing how a particle moves along a predefined path in the coordinate plane. This often necessitates examining parametric equations, which help us track the particle's position over time.
In the exercise given, the parametric equations \(x = t\) and \(y = \sqrt{4 - t^2}\) effectively describe the motion of the particle. Here, the parameter \(t\) represents time. As \(t\) changes from 0 to 2, the position \((x, y)\) of the particle changes accordingly.
To understand the direction and trace of the motion:

• The particle begins at the point \((x=0, y=2)\), when \(t = 0\).
• As \(t\) increases, the \(x\)-coordinate increases linearly, while the \(y\)-coordinate varies as a function of \(t\)
• Finally, at \(t = 2\), the particle reaches \((x=2, y=0)\).

The motion along the path, therefore, follows a rightward direction, illustrating the change from the starting to ending point as dictated by \(t\).

Graphing in the Coordinate Plane

Visualizing equations on the coordinate plane can significantly enhance comprehension and analysis. The graph gives us a spatial understanding of how two variables relate.
With the equation \(x^2 + y^2 = 4, y \geq 0\), plotting this on the plane involves drawing the upper half of a circle. The center of the circle is at the origin \((0,0)\), and the radius is 2. This can be visually confirmed as a semicircle that lies on the upper part of the circle extending horizontally from \((-2, 0)\) to \((2, 0)\).
One can quickly graph this by:

• Determining its shape based on the equation.
• Drawing points from \((0, 2)\) to \((2, 0)\).
• Ensuring the curve reflects the given limitations and conditions, such as applies only to \(y \geq 0\).

This method of visual representation makes it easier to interpret the movement and behavior of the particle as provided by its parametric descriptions.

Parametric to Cartesian Conversion

Converting parametric to Cartesian equations simplifies tracking relationships between two variables by removing the parameter and relating the variables directly.
In parametric forms, you set explicit paths using an additional variable, often serving as a function of time or another aspect. Transitioning to a Cartesian equation involves eliminating this third variable, allowing us to express \(y\) directly in terms of \(x\) without \(t\).
For the exercise, we start with:

• \(x = t\) \(\rightarrow x\) directly replaces \(t\).
• The subsequent equation \(y = \sqrt{4 - t^2}\)
• Transforms to \(y = \sqrt{4 - x^2}\)

This conversion tells us precisely how \(x\) and \(y\) relate without needing \(t\). The result is a more comprehensive equation, \(x^2 + y^2 = 4\), highlighting the upper semicircle's path. By converting, we transition from specifics of motion over a parameter to an overarching view of the relationship between \(x\) and \(y\), aiding in analysis and understanding.