Chapter 1.4, Problem 36E

(a)

To determine

To find: The parameterization to model the motion of a particle that starts at $\left(-a,0\right)$ and traces the circle ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$ , $a>0$ , $b>0$ once clockwise.

Answer to Problem 36E

The possible parameterization is $x=a\cos t$ and $y=-b\sin t$ in the interval $-\pi \le t\le \pi$.

Explanation of Solution

Given information: The particle starts moving from $\left(-a,0\right)$ and traces the circle ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$ , $a>0$ , $b>0$ once clockwise.

Calculation:

The standard parameterization for an ellipse with horizontal semi-axis $a$ and vertical axis $b$ is given by $x=a\cos t$ and $y=b\sin t$.

As this will cause to move the particle in counterclockwise so set $t=-t$.

Substitute $-t$ for $t$ in the equations $x=a\cos t$ and $y=b\sin t$.

  $\begin{array}{c}x=a\cos \left(-t\right)\\ =a\cos t\\ y=b\sin \left(-t\right)\\ =-b\sin t\end{array}$

Substitute $t=-\pi$ for the starting point.

  $\begin{array}{c}x=a\cos \left(-\pi \right)\\ =-a\\ y=-b\sin \left(-\pi \right)\\ =0\end{array}$

The particle will complete one cycle in an interval of $2\pi$ . So, substitute an end point at $-\pi +2\pi =\pi$.

  $\begin{array}{c}x=a\cos \left(-\pi \right)\\ =-a\\ y=b\sin \left(-\pi \right)\\ =0\end{array}$

Therefore, the possible parameterization is $x=a\cos t$ and $y=-b\sin t$ in the interval $-\pi \le t\le \pi$.

(b)

To determine

To find: The parameterization to model the motion of a particle that starts at $\left(-a,0\right)$ and traces the circle ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$ , $a>0$ , $b>0$ once counterclockwise.

Answer to Problem 36E

The possible parameterization is $x=a\cos t$ and $y=b\sin t$ in the interval $-\pi \le t\le \pi$.

Explanation of Solution

Given information: The particle starts moving from $\left(-a,0\right)$ and traces the circle ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$ , $a>0$ , $b>0$ once counterclockwise.

Calculation:

The standard parameterization for an ellipse with horizontal semi-axis $a$ and vertical axis $b$ is given by $x=a\cos t$ and $y=b\sin t$ by which the particle moves counterclockwise.

Substitute $t=-\pi$ for the starting point.

  $\begin{array}{c}x=a\cos \left(-\left(-\pi \right)\right)\\ =-a\\ y=b\sin \left(-\left(-\pi \right)\right)\\ =0\end{array}$

The particle will complete one cycle in an interval of $2\pi$ . So, substitute an end point at $-\pi +2\pi =\pi$.

  $\begin{array}{c}x=a\cos \left(-\pi \right)\\ =-a\\ y=b\sin \left(-\pi \right)\\ =0\end{array}$

Therefore, the possible parameterization is $x=a\cos t$ and $y=b\sin t$ in the interval $-\pi \le t\le \pi$.

(c)

To determine

To find: The parameterization to model the motion of a particle that starts at $\left(-a,0\right)$ and traces the circle ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$ , $a>0$ , $b>0$ twice clockwise.

Answer to Problem 36E

The possible parameterization is $x=a\cos t$ and $y=-b\sin t$ in the interval $-\pi \le t\le 3\pi$.

Explanation of Solution

Given information: The particle starts moving from $\left(-a,0\right)$ and traces the circle ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$ , $a>0$ , $b>0$ twice clockwise.

Calculation:

The standard parameterization for an ellipse with horizontal semi-axis $a$ and vertical axis $b$ is given by $x=a\cos t$ and $y=b\sin t$.

As this will cause to move the particle in counterclockwise so set $t=-t$.

Substitute $-t$ for $t$ in the equations $x=a\cos t$ and $y=b\sin t$.

  $\begin{array}{c}x=a\cos \left(-t\right)\\ =a\cos t\\ y=b\sin \left(-t\right)\\ =-b\sin t\end{array}$

Substitute $t=-\pi$ for the starting point.

  $\begin{array}{c}x=a\cos \left(-\pi \right)\\ =-a\\ y=-b\sin \left(-\pi \right)\\ =0\end{array}$

The particle will complete two cycle in an interval of $4\pi$ . So, substitute an end point at $-\pi +4\pi =3\pi$.

  $\begin{array}{c}x=a\cos \left(-3\pi \right)\\ =-a\\ y=b\sin \left(-3\pi \right)\\ =0\end{array}$

Therefore, the possible parameterization is $x=a\cos t$ and $y=-b\sin t$ in the interval $-\pi \le t\le 3\pi$.

(d)

To determine

To find: The parameterization to model the motion of a particle that starts at $\left(-a,0\right)$ and traces the circle ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$ , $a>0$ , $b>0$ twice counterclockwise.

Answer to Problem 36E

The possible parameterization is $x=a\cos t$ and $y=b\sin t$ with the interval $-\pi \le t\le 3\pi$.

Explanation of Solution

Given information: The particle starts moving from $\left(-a,0\right)$ and traces the circle ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$ , $a>0$ , $b>0$ twice counterclockwise.

Calculation:

The standard parameterization for an ellipse with horizontal semi-axis $a$ and vertical axis $b$ is given by $x=a\cos t$ and $y=b\sin t$ by which the particle moves counterclockwise.

Substitute $t=-\pi$ for the starting point.

  $\begin{array}{c}x=a\cos \left(-\left(-\pi \right)\right)\\ =-a\\ y=b\sin \left(-\left(-\pi \right)\right)\\ =0\end{array}$

The particle will complete one cycle in an interval of $4\pi$ . So, substitute an end point at $-\pi +4\pi =3\pi$.

  $\begin{array}{c}x=a\cos \left(3\pi \right)\\ =-a\\ y=b\sin \left(3\pi \right)\\ =0\end{array}$

Therefore, the possible parameterization is $x=a\cos t$ and $y=b\sin t$ in the interval $-\pi \le t\le 3\pi$.