Problem 71 - 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 t, \quad y=9 t^{2}, \quad-\infty

Step-by-Step Solution

Answer

The particle's path is a parabola, \( y = x^2 \), traced from left to right.

Step 1: Convert Parametric Equations to Cartesian Equation

To find the Cartesian equation, we need to eliminate the parameter \( t \) from the equations. We have \( x = 3t \) and \( y = 9t^2 \). Solve for \( t \) from \( x = 3t \): \( t = \frac{x}{3} \). Substitute this into \( y = 9t^2 \) to get \( y = 9 \left( \frac{x}{3} \right)^2 = y = 9 \cdot \frac{x^2}{9} = x^2 \).

Step 2: Write the Cartesian Equation

After substitution and simplification, we obtain the Cartesian equation of the curve as \( y = x^2 \). This represents a parabola opening upwards.

Step 3: Determine the Graph and Direction of Motion

The Cartesian equation \( y = x^2 \) is a standard parabola. The direction of motion is determined by observing the parameter \( t \). As \( t \) moves from \(-\infty\) to \(+\infty\), \( x = 3t \) implies \( x \) also covers its entire range from \(-\infty\) to \(+\infty\). This means the particle moves along the parabola from left to right as \( t \) increases.

Step 4: Graph the Cartesian Equation and Motion

Graph the parabola \( y = x^2 \) on the coordinate plane. Indicate the path of the particle with arrows moving from left to right along the parabola, as \( t \) changes from \(-\infty\) to \(+\infty\). The entire parabola is traced due to the given range of \( t \).

Key Concepts

Cartesian Equation

In mathematics, a Cartesian equation describes a relationship between the x and y coordinates in a plane, eliminating any parameter variables. This approach turns parametric equations into one overarching equation that captures the path of a curve on the Cartesian plane.
For example, certain motions in physics or geometry, such as a particle's path, might initially be expressed with parametric equations like \( x = 3t \) and \( y = 9t^2 \). Here, \( t \) serves as a parameter representing time, which can be eliminated to derive a clearer two-dimensional representation.

• Start with the first equation \( x = 3t \) to solve for the parameter: \( t = \frac{x}{3} \).
• Next, substitute \( t \) in the second equation, \( y = 9t^2 \), leading to \( y = 9(\frac{x}{3})^2 \), which simplifies to \( y = x^2 \).

Thus, the parametric equations are reduced to the straightforward Cartesian equation: \( y = x^2 \). This succinctly describes the path of the motion through a parabola on the Cartesian plane.

Particle Motion

Exploring particle motion involves understanding how an object's position changes over time, often modeled through equations with a time parameter like \( t \).
This kind of motion can initially be presented with parametric equations, providing separate expressions for position coordinates based on \( t \). For a particle traveling through the plane, the parametric equations \( x = 3t \) and \( y = 9t^2 \) define its precise location at any moment in time.

• As \( t \) progresses over its full range from \(-\infty \) to \(+\infty \), it influences \( x \) and \( y \) values according to these equations.
• This leads to the conclusion that the particle's path covers an endless range along the Cartesian plane.
• The key aspect is the particle's trajectory, represented by the equation \( y = x^2 \) in Cartesian coordinates.

The motion follows the parabola from left to right, signifying its continuous transition across the coordinate plane as time evolves.

Graphing Parabolas

A fundamental feature associated with parabolas is their standard form, often represented as \( y = ax^2 + bx + c \), where \( a, b, \) and \( c \) define the parabola's shape and position. In this problem, the parametric equations simplify to the Cartesian form \( y = x^2 \), emblematic of an upward-opening parabola.
A parabola appears as a U-shaped curve on the graph, symmetrical around its vertical axis.

• The vertex, positioned at the origin \((0,0)\) in this case, is the lowest point because the parabola opens upwards.
• To graph \( y = x^2 \), plot symmetric points on either side of \( x = 0 \), such as \( (-2,4), (-1,1), (0,0), (1,1), (2,4) \).
• Connect these points to form the parabola, ensuring it reflects the mathematical elegance of symmetry.

Furthermore, understanding the flow of particle motion within parametric conditions enhances the graph's interactivity. Arrows can be drawn along the parabola, indicating movement from left to right as \( t \) increases, embodying continuous and coherent motion. This dynamic aspect provides a holistic picture of the parabola's graph in the context of real-world particle motion.