Problem 19 - Calculus Early Transcendentals

Question

Describe the motion of a particle with position \((x, y)\) as \(t\) varies in the given interval. \(x=3+2 \cos t, \quad y=1+2 \sin t, \quad \pi / 2 \leqslant t \leqslant 3 \pi / 2\)

Step-by-Step Solution

Answer

The particle moves along a semicircle clockwise, from top \((3,3)\) to bottom \((3,-1)\).

Step 1: Identifying the Parametric Equations

The position of the particle is given by the parametric equations: \(x = 3 + 2\cos t\) and \(y = 1 + 2\sin t\). These equations determine the particle's path in the \((x, y)\) coordinate plane as the parameter \(t\) changes.

Step 2: Recognize the Shape of the Path

The equations \(x = 3 + 2\cos t\) and \(y = 1 + 2\sin t\) are the parametric equations of a circle with center \((3, 1)\) and radius 2. This is determined by the standard form \((x-h)^2+(y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 3: Determine the Interval of Motion

The interval for \(t\) is \(\pi/2 \leq t \leq 3\pi/2\). In terms of the unit circle in trigonometry, starting at \(t = \pi/2\) means starting at the top of the circle, moving downwards. Ending at \(t = 3\pi/2\) means reaching the bottom of the circle.

Step 4: Describe the Direction of Motion

As \(t\) increases from \(\pi/2\) to \(3\pi/2\), \(\cos t\) changes from 0 to -1 and then back to 0, while \(\sin t\) changes from 1 to 0 and then to -1. This indicates that the particle moves along the semicircle from the top (where \(y\) is maximum) going downward to the bottom (where \(y\) is minimum), moving clockwise.

Step 5: Summarize the Motion

The particle moves along a semicircular path. Starting at the point \((3, 3)\) when \(t = \pi/2\) on the top of the semicircle, the particle moves clockwise to the point \((3, -1)\) at \(t = 3\pi/2\).

Key Concepts

Particle Motion

In mathematics, particle motion refers to the way in which a particle moves through space over time. To understand this, we often use equations that describe the particle's position in terms of a variable, typically denoted as \( t \). In this context, the position of a particle is given by coordinates \( (x, y) \), indicating its location on a 2D plane.
For instance, parametric equations are an effective way to describe particle motion. These types of equations allow us to express \( x \) and \( y \) each as functions of \( t \), known as the parameter. By varying \( t \), we can track the precise location of the particle at any given moment.
Understanding particle motion helps us visualize how the particle moves along its path without the need for a traditional "function" form, thereby enriching our understanding of motion dynamics.

Parametric Equations in Calculus

Parametric equations provide a powerful tool in calculus, particularly when investigating motion. Unlike standard Cartesian equations that delineate direct relationships between \( x \) and \( y \), parametric equations allow us to express both \( x \) and \( y \) as separate functions of another variable, often \( t \).
In the given exercise, the equations \( x = 3 + 2\cos t \) and \( y = 1 + 2\sin t \) describe the trajectory of the particle. Here, \( t \) is a parameter that changes within a specified interval—in this case, from \( \pi/2 \) to \( 3\pi/2 \).
This system is particularly advantageous when modeling complex shapes or motions that are not easily expressed by \( y = f(x) \). It's useful to:

• Helps in understanding complex movement or curves.
• Separates the influence of time (or another variable) on each coordinate.
• Facilitates computations involving changes over time, like velocity or acceleration.

You can experiment with graphing these equations for various \( t \) values to see how the particle's path unfolds over time.

Trigonometric Functions in Motion

Trigonometric functions, such as sine and cosine, play a crucial role in modeling periodic and circular motion. In the realm of particle motion, these functions help frame movements that repeat at regular intervals, mimicking natural phenomena like the cycles of day and night or the oscillation of a pendulum.
For instance, in our exercise, \( x = 3 + 2\cos t \) and \( y = 1 + 2\sin t \) use cosine and sine functions to create a circular path. This is possible because:

• \( \cos t \) controls the horizontal movement (\( x \) direction) and cycles from -1 to 1.
• \( \sin t \) manages the vertical movement (\( y \) direction) and also cycles between -1 and 1.

As \( t \) progresses from \( \pi/2 \) to \( 3\pi/2 \), these trigonometric functions ensure that the particle follows a semicircular path. Observing these cycles is essential when predicting motion patterns in physics or engineering, where periodic motion is a common occurrence.