Jason Farmer

2020-10-27

That parametric equations
contain more information than just the shape of the
curve. Write a short paragraph explaining this
statement. Use the following example and your
answers to parts (a) and (b) below in your
explanation.
The position of a particle is given by the parametric
equations $x=\text{}\mathrm{sin}\text{}(t)\text{}and\text{}y=\text{}\mathrm{cos}\text{}(t)$ where t
represents time. We know that the shape of the path
of the particle is a circle.
a) How long does it take the particle to go once
around the circle? Find parametric equations if the
particle moves twice as fast around the circle.
b) Does the particle travel clockwise or
counterclockwise around the circle? Find parametric
equations if the particle moves in the opposite
direction around the circle.

delilnaT

Skilled2020-10-28Added 94 answers

Step 1
(a) Note that, the position of the particle is given by the parametric equations $x=\text{}\mathrm{sin}t,\text{}and\text{}y=\text{}\mathrm{cos}\text{}t$ .
The parametric equations contain more than just shape of the curve. They also represent the direction of curve as traveling. If a position of a particle is determined by the equation $x=\text{}\mathrm{sin}\text{}t,\text{}y=\text{}\mathrm{cos}\text{}t,$ this set of equations denotes which direction the particle is traveling based on different times t.
For example, at $t=0,\text{}\text{the particle is at the point}\text{}(0,\text{}1)\text{}\text{but at time}\text{}t=\text{}\frac{\pi}{2}\text{}\text{the particle has moved to the point}\text{}(1,\text{}0)$ in a clockwise direction
As the period of the parametric equations is $2\text{}\pi $ , to find for the particle to travel a full rotation around the circle.
It will take the time $t=2\text{}\pi $ to traverse the circle in a clockwise direction.
To travel the circle twice as fast simply double the coefficient inside each trigonometric function and the parametric equations are $x=\text{}\mathrm{sin}\text{}2t,\text{}y=\text{}\mathrm{cos}\text{}2t.$
Thus, the time that will be taken by the particle to go once around the circle is $t=2\text{}\pi \text{}\text{and the parametric equations, the particle moves twice as fast around the circle are}\text{}x=\text{}\mathrm{sin}\text{}2t,\text{}y=\text{}\mathrm{cos}\text{}2t.$
Step 2
(b) Note that, the particle travels clockwise.
For example, at $t=0,\text{}\text{the particle is at the point}\text{}(0,\text{}1),\text{}\text{but at the time}\text{}t=\text{}\frac{\pi}{2}\text{}\text{the particle has moved to the point}\text{}(1,\text{}0)$ in a clockwise direction.
The parametric equations when the particle travels in the opposite direction, the parametric equations will be exchanged.
That are, $x=\text{}\mathrm{cos}\text{}t,\text{}y=\text{}\mathrm{sin}\text{}t.$
Thus, the particle travels clockwise and if the particle travels in opposite direction around the circle, the parametric equations are $x=\text{}\mathrm{cos}\text{}t,\text{}y=\text{}\mathrm{sin}\text{}t.$

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