Problem 3 - 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. $$x=2 t-5, \quad y=4 t-7, \quad-\infty< t <\infty$$

Step-by-Step Solution

Answer

The particle moves along the line \( y = 2x + 3 \) from left to right.

Step 1: Understanding the Parametric Equations

We are given parametric equations: \( x = 2t - 5 \) and \( y = 4t - 7 \). These equations describe the coordinates \((x, y)\) of a particle as a function of the parameter \( t \). Our task is to find a Cartesian equation that eliminates \( t \), leaving a relationship between \( x \) and \( y \).

Step 2: Eliminating the Parameter

To eliminate \( t \), first solve the equation for \( x \): \( x = 2t - 5 \). Rearranging gives \( t = \frac{x + 5}{2} \). Substitute this expression for \( t \) into the equation for \( y \): \[y = 4\left(\frac{x + 5}{2}\right) - 7\]Simplify this to find the Cartesian equation.

Step 3: Simplifying the Cartesian Equation

Substitute \( t = \frac{x + 5}{2} \) into the \( y \) equation:\[y = 4\left(\frac{x + 5}{2}\right) - 7 = 2(x + 5) - 7\]Distribute and simplify further:\[y = 2x + 10 - 7 = 2x + 3\]This is the Cartesian equation of the path the particle takes.

Step 4: Analyzing the Cartesian Equation

The Cartesian equation \( y = 2x + 3 \) is a linear equation representing a straight line. This equation suggests the particle moves along a straight path with a slope of 2 and a y-intercept of 3.

Step 5: Graphing the Cartesian Equation

Graph the line \( y = 2x + 3 \). The line has a positive slope of 2, meaning it rises 2 units in the \( y\)-direction for every 1 unit it moves to the right in the \( x\)-direction. Mark the y-intercept at \( (0, 3) \) and trace the line accordingly.

Step 6: Determining Motion and Direction

As \( t \) increases, both \( x \) and \( y \) increase. This indicates that the particle moves from left to right along the line \( y = 2x + 3 \). The full range of the line is the path of the particle because \(-\infty < t < \infty\).

Key Concepts

Cartesian Equation

Cartesian equations provide a relationship between two variables, usually denoted as \(x\) and \(y\), without involving a parameter like \(t\). For the particle path problem given, we started with parametric equations: \(x = 2t - 5\) and \(y = 4t - 7\). These equations depend on the parameter \(t\), which represents time or another independent variable.

To convert these parametric equations into a single Cartesian equation, we need to eliminate \(t\). This involves solving one of the parametric equations for \(t\) and substituting that expression into the other equation. In our example, solving \(x = 2t - 5\) for \(t\) resulted in \(t = \frac{x + 5}{2}\). By substituting this \(t\) into the equation for \(y\), we find that \(y = 2x + 3\).

• The substitution allows us to express \(y\) directly in terms of \(x\), leading to a Cartesian equation.
• This equation, \(y = 2x + 3\), describes a line in the \(xy\)-plane.

Particle Motion

The concept of particle motion in this context refers to the movement of a particle along a defined path in the plane. The parametric equations \(x = 2t - 5\) and \(y = 4t - 7\) describe the trajectory of the particle as the parameter \(t\) varies.

- As \(t\) increases, the equations show how the \(x\) and \(y\) coordinates of the particle change over time. The change in \(x\) expressed by \(2t - 5\) and in \(y\) by \(4t - 7\) implies a constant rate of change for both coordinates.- This constant rate gives rise to linear motion, as the particle moves in a straight line.

The full range of the parameter \(t\) from \(-\infty\) to \(\infty\), ensures the particle can move indefinitely in both directions, implying there are no restrictions on the path. In this case, the entire line \(y = 2x + 3\) is the path of the particle.

- Movement is continuous with no interruptions along the line, and since the particle is not constrained by bounds on \(t\), the potential path spans the entire Cartesian line.

Graphing Linear Equations

Graphing linear equations helps visualize mathematical relationships and solutions clearly. To graph the linear equation \(y = 2x + 3\), start by identifying key characteristics such as slope and intercepts.

- The slope of \(2\) indicates how steep the line is. It rises \(2\) units vertically for every \(1\) unit it moves horizontally. This shows the direction and rate of the particle's movement.
- The y-intercept is the point where the line crosses the y-axis. For \(y = 2x + 3\), this occurs at the point \((0, 3)\).

• Begin the graph by plotting the y-intercept (0, 3).
• From there, use the slope to determine another point on the line, such as moving up 2 units and right 1 unit to define a next point.

Connect these points extending the line in both directions to represent that the particle's path continues indefinitely along this line.

- With the graph of \(y = 2x + 3\), it becomes easy to see the linear trajectory of the particle and understand how different values of \(x\) relate to \(y\). The graph helps in visualizing direction and extent of the path, aiding in deeper understanding of particle motion.