Problem 20 - Thomas Calculus

Question

Find parametric equations and a parameter interval for the motion of a particle that starts at \((a, 0)\) and traces the ellipse \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1\) \(\begin{array}{ll}{\text { a. once clockwise. }} & {\text { b. once counterclockwise. }} \\ {\text { c. twice clockwise. }} & {\text { d. twice counterclockwise. }}\end{array}\) (As in Exercise \(19,\) there are many correct answers.)

Step-by-Step Solution

Answer

a. \([0, 2\pi]\) clockwise; b. \([0, 2\pi]\) counterclockwise; c. \([0, 4\pi]\) clockwise; d. \([0, 4\pi]\) counterclockwise.

Step 1: Parametric Equation Formulation for an Ellipse

To parameterize the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we use parametric equations for ellipses. The generic parametric form is:\[ x(t) = a \cos(t) \]\[ y(t) = b \sin(t) \]Here, \( t \) is the parameter, typically representing an angle in radians.

Step 2: Parametric Equations for Clockwise Motion

For motion in the clockwise direction, we need to reverse the direction of parameter increase. We can achieve clockwise motion by negating the parameter function for \( y(t) \):\[ x(t) = a \cos(t) \]\[ y(t) = -b \sin(t) \]

Step 3: Clockwise Parametric Equations - Interval for Once Around

To travel around the ellipse once in the clockwise direction, \( t \) needs to go from \( 0 \) to \( 2\pi \). Thus, the parameter interval is \([0, 2\pi]\).

Step 4: Parametric Equations for Counterclockwise Motion

For counterclockwise motion, use the standard parameter functions:\[ x(t) = a \cos(t) \]\[ y(t) = b \sin(t) \]

Step 5: Counterclockwise Parametric Equations - Interval for Once Around

To move once around the ellipse counterclockwise, \( t \) should also range from \( 0 \) to \( 2\pi \). So, the parameter interval is \([0, 2\pi]\).

Step 6: Parametric Equations for Twice Around (Clockwise)

To travel around the ellipse twice in the clockwise direction, keep the clockwise equations and let \( t \) go from \( 0 \) to \( 4\pi \) to complete two full rotations:\[ x(t) = a \cos(t) \]\[ y(t) = -b \sin(t) \]The interval is \([0, 4\pi]\).

Step 7: Parametric Equations for Twice Around (Counterclockwise)

For twice around the ellipse counterclockwise, maintain the standard equations and extend the interval:\[ x(t) = a \cos(t) \]\[ y(t) = b \sin(t) \]The interval is \([0, 4\pi]\) to complete two full counterclockwise cycles.

Key Concepts

ellipse

An ellipse is a shape that can be thought of as a squished circle. It stretches along two main directions, which are defined by its semi-major and semi-minor axes. The standard equation of an ellipse is given by the formula:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where:

• \( a \) is the semi-major axis, representing the ellipse's stretch along the x-direction.
• \( b \) is the semi-minor axis, showing the y-direction stretch.

This means if you plug different values for \( a \) and \( b \), the ellipse will look different – more stretched or squashed – but it will always remain symmetrical by design.Understanding this concept of ellipses is crucial when working with parametric equations, which help in describing such geometric curves with a parameter, rather than a direct relationship between x and y.

parametric motion

Parametric motion describes how you can represent complex paths and behaviors in mathematics using parameters. Instead of thinking about a point's coordinates directly, parametric equations introduce a new variable, usually \( t \), that calculates x and y coordinates as functions of this parameter.For ellipses:

• The x-coordinate is represented as \( x(t) = a \cos(t) \).
• The y-coordinate is described as \( y(t) = b \sin(t) \).

These equations let us visualize motion along the ellipse as \( t \) changes. This is especially useful in physics and engineering, where movements and paths are often described using time or angles. Parametric equations are powerful tools for modeling not just ellipses, but any kind of curve or motion.

clockwise and counterclockwise

When tracing an ellipse, understanding the direction of movement – clockwise or counterclockwise – is important. This direction changes based on how you define the parameter equations.To trace an ellipse clockwise:

• Use \( x(t) = a \cos(t) \).
• Use \( y(t) = -b \sin(t) \).

The negative sign in the \( y(t) \) equation causes the motion to reverse as \( t \) increases, creating a clockwise movement.For a counterclockwise path:

• Use \( x(t) = a \cos(t) \).
• Use \( y(t) = b \sin(t) \).

Here, both equations maintain the natural increase of the parameter, resulting in a counterclockwise trajectory. These directions are essential for tasks in animation or any process where the direction of motion needs to be precisely controlled.