Problem 75 - Thomas Calculus

Question

Exercises \(67-78\) 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. $$ x=t, \quad y=\sqrt{1-t^{2}}, \quad-1 \leq t \leq 0 $$

Step-by-Step Solution

Answer

The path is the upper semi-circle, left side, with direction from \(x = -1\) to \(x = 0\).

Step 1: Understanding the Parametric Equations

The given parametric equations are: \[ x = t \] \[ y = \sqrt{1-t^2} \] and the parameter interval is \(-1 \leq t \leq 0\). These equations describe how the coordinates \((x, y)\) change with parameter \(t\) in the xy-plane.

Step 2: Eliminate the Parameter

To find a Cartesian equation, eliminate the parameter \(t\) from the parametric equations. From \(x = t\), we have \(t = x\). Substitute this value into the equation for \(y\):\[ y = \sqrt{1-x^2} \].

Step 3: Determine the Cartesian Equation

The resulting equation \(y = \sqrt{1-x^2} \) is the Cartesian equation of the path traced by the particle. This equation represents the upper half of a circle with radius 1, centered at the origin.

Step 4: Consider the Parameter Interval

The parameter interval \(-1 \leq t \leq 0\) translates to \(-1 \leq x \leq 0\) due to the equation \(x = t\). This interval indicates that the particle travels along the left semi-circle.

Step 5: Identify Direction of Motion

Since the parameter \(t\) increases from \(-1\) to \(0\), the motion starts at \(x = -1\) and progresses towards \(x = 0\). Thus, the particle moves from left to right along the path.

Step 6: Graph the Cartesian Equation

Graph the equation \( y = \sqrt{1-x^2} \) for \( -1 \leq x \leq 0 \). This portion of the graph is the left half of the upper semi-circle with radius 1 centered at the origin. Draw an arrow along this path to indicate the direction from \(x = -1\) to \(x = 0\).

Key Concepts

Cartesian Equation

When we talk about converting parametric equations into a Cartesian equation, we're looking at a way to express a curve without relying on parameters such as \( t \). With parametric equations, both \( x \) and \( y \) are defined in terms of \( t \), like in our case where \( x = t \) and \( y = \sqrt{1-t^2} \). However, to find the Cartesian equation, we need to eliminate \( t \).Here's how it's done:

• Since \( x = t \), we can substitute \( t \) with \( x \) in the other equation: \( y = \sqrt{1-x^2} \).
• This resulting function, \( y = \sqrt{1-x^2} \), is now a Cartesian equation.

This Cartesian equation describes the path of the particle without needing to refer back to the parameter \( t \). It represents a geometric shape, in this case, part of a circle.

Particle Path

A particle's path is essentially the trajectory that the particle follows as \( t \) changes. In this problem, we discovered that the Cartesian equation \( y = \sqrt{1-x^2} \) represents the upper portion of a circle with a radius of 1, centered at the origin.Since our parametric equations lead to this Cartesian form:

• The path can be visualized as the particle moving along the arc of this circle, specifically the upper half.

The interval \(-1 \leq t \leq 0\) ensures that the particle only covers the left half of this arc, moving from the leftmost point \((-1, 0)\) to the origin \((0, 1)\). The circle cut-off from \(x = 0\) makes the path clearer to imagine as a portion of the circle rather than the complete circumference.

Graphing

Graphing is a crucial part of interpreting mathematical equations visually. For our situation with the equation \( y = \sqrt{1-x^2} \), graphing it means plotting points on the coordinate plane to show the actual path the particle takes.Here's a step-by-step guide:

• Draw the standard coordinate axes.
• Plot the equation \( y = \sqrt{1-x^2} \) within the given interval \(-1 \leq x \leq 0\).
• This section of the graph will trace a curved line from \( (-1, 0) \) to the origin \((0, 1)\).
• Add an arrow or mark the direction starting from \((-1, 0)\) to \((0, 1)\) to indicate the direction of motion of the particle.

By graphing these points, you effectively visualize the particle's path and its motion during the specified interval.

Parameter Interval

The parameter interval is a critical component because it tells us which portion of the Cartesian path the particle travels. In our case, the interval is \(-1 \leq t \leq 0\).This translates directly to the same interval for \(x\) because \( x = t \):

• This limitation defines the scope of \(x\) values, confining them between -1 and 0, and thereby shaping the segment of the circle that is considered.

This interval allows us to depict how far and in what direction our particle travels.

• The interval begins at \(t = -1\) corresponding to \((-1, 0)\) in Cartesian coordinates and ends at \(t = 0\), corresponding to \((0, 1)\).
• The motion traces from left to right along the defined quarter-circle arc.

Understanding parameter intervals is key as they define the start and endpoint of motion, establishing the full pathway followed on the curve.