Problem 5 - 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. \(x=t, \quad y=\sqrt{t} ; \quad t \geq 0\)

Step-by-Step Solution

Answer

Path is \( y = \sqrt{x} \), traced from (0,0) to the right, starting as \( t=0 \).

Step 1: Start with Parametric Equations

The parametric equations of motion are given as \( x = t \) and \( y = \sqrt{t} \). The parameter interval is \( t \geq 0 \). This means for every non-negative value of \( t \), we can find corresponding values for \( x \) and \( y \).

Step 2: Find the Cartesian Equation

To find the Cartesian equation, we need to eliminate the parameter \( t \) from the parametric equations. From \( x = t \), we get \( t = x \). Substituting this into \( y = \sqrt{t} \), we get \( y = \sqrt{x} \). The Cartesian equation is therefore \( y = \sqrt{x} \).

Step 3: Graph the Cartesian Equation

The Cartesian equation \( y = \sqrt{x} \) represents the right half of a parabola opening to the right. To graph it, plot points for non-negative values of \( x \) and plot the corresponding \( y \) values. The graph starts at the origin (0,0) and extends to the right and upwards (since \( y = \sqrt{x} \) is defined only for \( x \geq 0 \)).

Step 4: Indicate the Path Traced by the Particle and Motion Direction

As \( t \) increases from 0 to infinity, the particle moves along the path of the parabola \( y = \sqrt{x} \), starting at the origin and moving upwards to the right. The direction of motion is from the origin along the curve towards increasing \( x \) and \( y \).

Key Concepts

Cartesian Equation

Parametric equations like \( x = t \) and \( y = \sqrt{t} \) describe motion using a parameter \( t \). To find the Cartesian equation of the particle's path, we need to eliminate this parameter. Given \( x = t \), we can substitute it into the second equation to express \( y \) in terms of \( x \).
This substitution results in the Cartesian equation \( y = \sqrt{x} \). This new form, where \( y \) is a function of \( x \), provides a simpler way of visualizing and analyzing the path on the Cartesian plane.
Cartesian equations are especially useful because they help in understanding the shape and nature of the curve traced by the particle without dealing with the parameter.

Graphing Parabolas

The Cartesian equation \( y = \sqrt{x} \) can be graphed to show the particle's path. This equation describes a parabola opening rightwards, capturing the path of the particle in a visual manner.
When graphing, it is helpful to identify key points. For example:

• At \( x = 0 \), \( y = \sqrt{0} = 0 \), giving the point (0, 0).
• At \( x = 1 \), \( y = \sqrt{1} = 1 \), giving the point (1, 1).
• At \( x = 4 \), \( y = \sqrt{4} = 2 \), giving the point (4, 2).

Plotting these points and smoothly connecting them illustrates how the parabola extends to the right. This graphical representation helps students see how the equation forms its characteristic shape and how it is covered by the particle's motion.

Particle Motion

Understanding particle motion through parametric equations involves recognizing how the parameter \( t \) drives movement. Here, as \( t \) increases:

• The \( x \)-coordinate increases since \( x = t \).
• The \( y \)-coordinate increases as \( y = \sqrt{t} \), but more slowly than \( x \).

This means that as the particle moves in time, it traces the curve portion of the parabola \( y = \sqrt{x} \) in the Cartesian plane.
The specific sequence of points and their connection forms a trajectory or path, depicting how the particle behaves over its range of motion. The path begins at the origin (0,0) and progresses to greater \( x \) and \( y \) values, visualizing the particle's continuous journey.

Parameter Interval

The parameter interval, \( t \geq 0 \), is crucial in defining the part of the path traced by the particle. This interval tells us that the behavior and position of the particle only make sense for non-negative values of \( t \).
By looking at the interval, we can conclude:

• The particle starts its journey from \( t = 0 \) onwards.
• There is no motion or position defined for \( t < 0 \).

The interval affects how we interpret the graph and set up initial conditions for studying the motion. Knowing the parameter interval helps in setting the boundaries for plotting points on the graph, making sure that we observe only the relevant part of the parabola produced by the parameter range.