Problem 73 - 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. $$ x=2 t-5, \quad y=4 t-7, \quad-\infty< t <\infty $$

Step-by-Step Solution

Answer

The path is a line given by the Cartesian equation \( y = 2x + 3 \), traced entirely from left to right.

Step 1: Write Parametric Equations

We are given the parametric equations for a particle's motion: \[ x = 2t - 5 \] \[ y = 4t - 7 \] The parameter interval given is \(-\infty < t < \infty\). Next, we will eliminate the parameter \(t\) to find the Cartesian equation of the path.

Step 2: Solve for t from the x equation

From the equation \( x = 2t - 5 \), solve for \( t \):\[ 2t = x + 5 \] \[ t = \frac{x + 5}{2} \]

Step 3: Substitute t in the y equation

Substitute \( t = \frac{x + 5}{2} \) into the equation for \( y \):\[ y = 4 \left( \frac{x + 5}{2} \right) - 7 \] Simplify the expression step by step:\[ y = 2(x + 5) - 7 \] \[ y = 2x + 10 - 7 \] \[ y = 2x + 3 \] This is the Cartesian equation of the path.

Step 4: Identify the Graph and Trace

The Cartesian equation \( y = 2x + 3 \) is a line. Since \( -\infty < t < \infty \), every point on the line is traced by the particle as \( t \) changes. The direction of motion can be found by observing the increase in \( t \): the path is traced from left to right (as \( x \) increases).

Step 5: Sketch the Graph

Sketch the graph of the line \( y = 2x + 3 \). Notice that it's a straight line with a slope of 2 and y-intercept 3. Since \(-\infty < t < \infty\), the particle traces the entire line, starting from left to right as \( t \) varies from \(-\infty\) to \(\infty\). Indicate arrows on the line to denote the direction of motion.

Key Concepts

Eliminating the Parameter

Parametric equations are a powerful way to describe the motion of particles, but sometimes it is useful to transform these back into a Cartesian equation. This process is known as "eliminating the parameter." First, let's understand the basic steps to eliminate a parameter like in this exercise. We begin by looking at the parametric equations: - \( x = 2t - 5 \) - \( y = 4t - 7 \)To eliminate the parameter \( t \), we solve one of the equations for \( t \). Here, using \( x = 2t - 5 \), we rearrange to find \( t \) in terms of \( x \): - \( t = \frac{x + 5}{2} \)Next, substitute this expression for \( t \) into the equation for \( y \). The substitution step links the two equations, eliminating \( t \) completely: - \( y = 4 \left( \frac{x + 5}{2} \right) - 7 \)Simplifying this expression results in - \( y = 2x + 3 \). The parameter \( t \) has successfully been eliminated, leading to a simpler expression describing the motion.

Cartesian Equation

Once the parameter \( t \) is eliminated, we find ourselves with a Cartesian equation. This is a simple mathematical relation between \( x \) and \( y \) that describes the path of the particle.In our example, the transformation resulted in the equation: - \( y = 2x + 3 \).This equation is in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here: - The slope \( m = 2 \) - The intercept \( b = 3 \)What does this mean in context? Beyond showing the path, it delivers specific characteristics at a glance:- The line has a moderate upward slope of 2, meaning it rises 2 units for every unit it moves to the right. - Starts at a y-value of 3 when \( x = 0 \).This Cartesian equation, thus, offers a neat, continuous description of the path the particle follows within the plane.

Particle Motion

Understanding "particle motion" within the context of parametric equations involves tracking how a particle moves along a path in two-dimensional space. The physical interpretation of these motions can be done through the Cartesian equation found during parameter elimination.Here, the equation \( y = 2x + 3 \) represents a line on the Cartesian plane. Because the parameter \( t \) varies from - \(-\infty \text{ to } \infty\), the particle travels over the entire line. Direction of travel is also essential to describe. We found that as \( t \) increases, both \( x \) and \( y \) increase, tracing the path from left to right. Practically: - Time is treated as moving, leading to the path being observed from one end to the other.Visualizing this path requires sketching the line, showing how: - The entire graph is traced by the particle. - Arrows indicate the direction, emphasizing the increase of \( t \).Engaging with particle motion through this method allows for an easy understanding of dynamic systems, traced smoothly over Cartesian planes.