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Chapter 9: Problem 18

\(15-18=\) Describe the motion of a particle with position \((x, y)\) as \(t\) varies in the given interval. \(x=\sin t, \quad y=\cos ^{2} t, \quad-2 \pi \leqslant t \leqslant 2 \pi\)

Short Answer

Expert verified
The particle oscillates in a wave pattern between \(x = -1\) and \(x = 1\), and between \(y = 0\) and \(y = 1\).

Step by step solution

01

Understand the Parametric Equations

The position of the particle is described by the parametric equations \(x = \sin t \) and \(y = \cos^2 t\). The parameter \(t\) ranges from \(-2\pi\) to \(2\pi\). Here, \(x\) represents the horizontal position, and \(y\) represents the vertical position of the particle.
02

Analyze the Range of Motion for \(x\)

The equation \(x = \sin t\) varies between -1 and 1 for \(t\) in the range \(-2\pi \leq t \leq 2\pi\). Therefore, the horizontal position \(x\) will oscillate back and forth over this section of the interval.
03

Analyze the Range of Motion for \(y\)

The equation \(y = \cos^2 t\) will vary between 0 and 1 because the square of the cosine function is always non-negative. For \(t\) in the range \(-2\pi \leq t \leq 2\pi\), \(\cos t\) swings between -1 and 1, causing \(y\) to oscillate fully between 0 and 1.
04

Determine the trajectory path

The trajectory is the set of points \((x, y)\) in the plane as \(t\) varies. Since \(x = \sin t\) and \(y = \cos^2 t\), \(x^2 + y = 1\) can be derived. This determines the path of the particle, forming a curve within the rectangle defined by \(-1 \leq x \leq 1\) and \(0 \leq y \leq 1\).
05

Describe the Motion

As \(t\) goes from \(-2\pi\) to \(2\pi\), the particle starts at \((0,1)\), moves right to \((1,0)\), left to \( (0,1) \), and then moves further left to \((-1,0)\), and repeats this motion back to \((0,1)\). It continues this oscillation back and forth on the x-axis from -1 to 1, while oscillating on the y-axis between 0 and 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Particle Motion
In this context, understanding particle motion involves looking at how a particle moves in a plane, with its position described using parametric equations. These equations define the x and y coordinates of the particle's position as functions of a third parameter, often time (denoted as \(t\)). In the given problem, the parametric equations are \(x = \sin t \) and \(y = \cos^2 t\). As \(t\) varies from \(-2\pi\) to \(2\pi\), these equations describe the trajectory of the particle over time.

Key Points about Particle Motion:
  • Position Coordinates: The x-coordinate, \(x = \sin t\), oscillates between -1 and 1. The y-coordinate, \(y = \cos^2 t\), oscillates between 0 and 1, creating a pattern of motion constrained within a rectangle when plotted on a coordinate plane.
  • Trajectories: The path the particle follows - called the trajectory - is determined by the combination of these oscillations, resulting in predictable repetitive movement.
  • Starting and Ending Points: As specified, the motion begins at \((0,1)\), proceeds in a loop through points like \((1, 0)\), and moves back to \((-1, 0)\), repeating in this pattern, which indicates a cyclical path.
Parametric Curves
Parametric curves are important for representing motion where standard \(y\) as a function of \(x\) graphs do not apply. In this exercise, we use parametric equations to plot the position of a moving particle through the plane. Parametric curves give us a powerful means to describe complex trajectories that aren't simple functions of each other.

Understanding the given Curve:
  • Equation Form: The relationship \(x^2 + y = 1\) emerges from our parameters, mixing both \(x\) and \(y\) in defining the path of our curve.
  • Bounding Box: The particle moves within the confining rectangle with corners defined by the limits \,\(-1 \leq x \leq 1\) and \,\(0 \leq y \leq 1\).
  • Visual Representation: Graphing these parametric equations results in a curve, which can look like parts of circles or ovals; they often form wave-like patterns across the plane, showing periodic motion.
Trigonometric Functions
Trigonometric functions are crucial in defining the parametric equations that describe the particle's motion. Specifically, \(\sin t\) and \(\cos^2 t\) are used to describe the particle's x and y positions over time.

Role of Trigonometric Functions:
  • Oscillation and Periodicity: Trigonometric functions are periodic, meaning they repeat their values in regular intervals. This periodicity defines the cyclical nature of the particle's motion in our problem.
  • Behavior and Range: The function \(\sin t\) ranges from -1 to 1, while \(\cos^2 t\) sits always between 0 and 1 due to squaring a cosine function (where \(-1 \leq \cos t \leq 1\)). This ensures non-negative y values.
  • Graphical Influence: These functions generate the specific patterns when graphed, creating paths and waves that define the trajectory of any particle modeled by such parametric functions.

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Most popular questions from this chapter

Graph the curve and find its length. $$x=\cos t+\ln \left(\tan \frac{1}{2} t\right), \quad y=\sin t, \quad \pi / 4 \leqslant t \leqslant 3 \pi / 4$$

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