Problem 74 - 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. $$ x=3-3 t, \quad y=2 t, \quad 0 \leq t \leq 1 $$

Step-by-Step Solution

Answer

The path is a line segment from (3,0) to (0,2), moving from right to left: \( x = 3 - \frac{3y}{2} \).

Step 1: Express y in terms of t

We are given the parametric equations: \( x = 3 - 3t \) and \( y = 2t \). First, solve for \( t \) using the equation for \( y \): \( t = \frac{y}{2} \).

Step 2: Substitute expression for t into x equation

Substitute \( t = \frac{y}{2} \) from the first step into the \( x \) equation to express \( x \) in terms of \( y \): \( x = 3 - 3\left(\frac{y}{2}\right) \).

Step 3: Simplify the expression

Simplify the equation from the previous step to find a relationship between \( x \) and \( y \): \( x = 3 - \frac{3y}{2} \). This is the Cartesian equation of the path.

Step 4: Find the range of values for y

Consider the parameter interval \( 0 \leq t \leq 1 \). For \( y = 2t \), substituting the limits gives \( 0 \leq y \leq 2 \).

Step 5: Determine x and y coordinates for given t

Calculate the coordinates of the path's start and endpoints: \( t = 0 \to (3, 0) \) and \( t = 1 \to (0, 2) \).

Step 6: Indicate direction of motion

The direction of motion follows the increase of \( t \) from \( (3, 0) \) to \( (0, 2) \). This means the path is traced from right to left along the line.

Key Concepts

Cartesian Equation

In the study of parametric equations, transforming them into a Cartesian equation can often simplify the analysis of a particle's path. The original parametric form given in the exercise is \( x = 3 - 3t \) and \( y = 2t \). To find the Cartesian equation, you need to eliminate the parameter \( t \) by expressing one variable in terms of the other.

Start by isolating \( t \) in the equation \( y = 2t \), which allows us to rewrite it as \( t = \frac{y}{2} \). Substitute this expression for \( t \) back into the equation for \( x \): \( x = 3 - 3\left(\frac{y}{2}\right) \).
This simplifies to the Cartesian equation \( x = 3 - \frac{3y}{2} \). This equation describes the relationship between \( x \) and \( y \) without any involvement of \( t \), providing a clear path to graphing the motion.

Particle Motion

Particle motion describes how a particle moves through the plane over time. In this exercise, the particle starts its motion at the point \( (3, 0) \) and ends at the point \( (0, 2) \). By using the value of \( t \), the parameter governing time, you can easily trace the path of the particle within the \(xy\)-plane.

The motion proceeds in a linear path because both parametric equations involve linear expressions in \( t \). As \( t \) increases from 0 to 1, \( x \) decreases from 3 to 0, while \( y \) increases from 0 to 2 linearly. This incremental change in position highlights the directional movement of the particle along the path described by the Cartesian equation \( x = 3 - \frac{3y}{2} \).

• At \( t = 0 \), the coordinates are \( (3, 0) \).
• As \( t = 1 \), the coordinates change to \( (0, 2) \).

This information specifies how the particle traverses over time.

Parameter Interval

In parametric equations, a parameter interval defines the specific range over which the parameter \( t \) varies, impacting how much of the path is traced by the particle. For this exercise, the interval \( 0 \leq t \leq 1 \) indicates the span of time during which the particle moves from its starting to its ending point.

Given this interval, the range for the \( y \)-values can be determined by substituting the smallest and largest values for \( t \) into \( y = 2t \). Thus, when \( t = 0 \), \( y = 0 \), and when \( t = 1 \), \( y = 2 \). Therefore, the \( y \)-values range from 0 to 2.

By considering the interval \( [0, 2] \) for \( y \) and \( [0, 3] \) for \( x \), it becomes apparent what segment of the line \( x = 3 - \frac{3y}{2} \) is traced by the particle. It is essential to keep in mind these limits when graphing, to ensure we showcase only the relevant portion of the graph influenced by the specified time interval.

Graphing in Mathematics

Graphing is a fundamental affair in mathematics when it comes to visualizing the motion of particles described by parametric equations. In this problem, we have converted a set of parametric equations into a Cartesian equation \( x = 3 - \frac{3y}{2} \). This linear equation represents a straight line in the Cartesian coordinate system.

When graphing this equation, it is crucial to mark the segment between the start point \( (3, 0) \) and the end point \( (0, 2) \), as defined by the parameter interval.

• The graph begins at \( (3, 0) \) and proceeds to \( (0, 2) \).
• The directional movement, from right to left, should be clearly indicated.

Utilizing the derived Cartesian equation, plotting the line becomes intuitive. By plotting the calculated points and connecting them with a line segment, you'll vividly demonstrate the motion of the particle over time.