Chapter 0.4, Problem 36E

(a)

To determine

To calculate : The parameterizations to model the motion of a particle once clockwise.

Answer to Problem 36E

The parameterizations to model the motion of a particle that starts at $\left(-a,0\right)$ and traces the ellipse ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1,a>0,b>0$ once clockwise is $x=-a\cos t,y=b\sin t,0\le t\le 2\pi$ .

Explanation of Solution

Given information :

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

Calculation:

For once clockwise motion of the particle time $0\le t\le 2\pi$ .

And particle should be at $\left(0,b\right)$ at $t=\frac{\pi }{2}$ and $\left(a,0\right)$ at $t=\pi$ since the starting point is $\left(-a,0\right)$ .

Therefore, the parametric equations are $x=-a\cos t,y=b\sin t,0\le t\le 2\pi$ .

(b)

To determine

To calculate : The parameterizations to model the motion of a particle once counter clockwise.

Answer to Problem 36E

The parameterizations to model the motion of a particle that starts at $\left(-a,0\right)$ and traces the ellipse ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1,a>0,b>0$ once counter clockwise is $x=-a\cos t,y=-b\sin t,0\le t\le 2\pi$ .

Explanation of Solution

Given information :

The motion of a particle that starts at $\left(-a,0\right)$ and traces the ellipse ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1,a>0,b>0$ once counter clockwise.

Calculation:

For once clockwise motion of the particle time $0\le t\le 2\pi$ .

And particle should be at $\left(0,-b\right)$ at $t=\frac{\pi }{2}$ and $\left(a,0\right)$ at $t=\pi$ since the starting point is $\left(-a,0\right)$ .

Therefore, the parametric equations are $x=-a\cos t,y=-b\sin t,0\le t\le 2\pi$ .

(c)

To determine

To calculate : The parameterizations to model the motion of a particle twice clockwise.

Answer to Problem 36E

The parameterizations to model the motion of a particle that starts at $\left(-a,0\right)$ and traces the ellipse ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1,a>0,b>0$ twice clockwise is $x=-a\cos t,y=b\sin t,0\le t\le 4\pi$ .

Explanation of Solution

Given information :

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

Calculation:

For twice clockwise motion of the particle time $0\le t\le 4\pi$ .

This can be achieved by doubling the time duration in parametric equations derived for clockwise.

Therefore, the parametric equations are $x=-a\cos t,y=b\sin t,0\le t\le 4\pi$ .

(d)

To determine

To calculate : The parameterizations to model the motion of a particle twice counter clockwise.

Answer to Problem 36E

The parameterizations to model the motion of a particle that starts at $\left(-a,0\right)$ and traces the ellipse ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1,a>0,b>0$ twice counter clockwise is $x=-a\cos t,y=-b\sin t,0\le t\le 4\pi$ .

Explanation of Solution

Given information :

The motion of a particle that starts at $\left(-a,0\right)$ and traces the ellipse ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1,a>0,b>0$ twice counter clockwise.

Calculation:

For twice clockwise motion of the particle time $0\le t\le 4\pi$ .

This can be achieved by doubling the time duration in parametric equations derived for clockwise.

Therefore, the parametric equations are $x=-a\cos t,y=b\sin t,0\le t\le 4\pi$ .