Problem 68
Question
More Information in Parametric Equations In this section we stated that parametric equations contain more information than just the shape of a 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 $$ x=\sin t \quad y=\cos t $$ 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.
Step-by-Step Solution
Verified Answer
Parametric equations show duration and direction of motion, beyond just shape.
1Step 1: Determine the Period of the Motion
The parametric equations \(x = \sin t\) and \(y = \cos t\) describe a circle. To determine how long it takes for the particle to complete one full circle, we find the period of these functions. Since \(\sin t\) and \(\cos t\) both have a fundamental period of \(2\pi\), it takes \(2\pi\) units of time for the particle to travel around the circle once.
2Step 2: Parametric Equations for Twice the Speed
To make the particle move twice as fast around the circle, we need to double its angular speed. We achieve this by replacing \(t\) with \(2t\). Therefore, the new parametric equations become \(x = \sin(2t)\) and \(y = \cos(2t)\). These equations describe a motion with a period of \(\pi\) because now the particle completes one full circle in half the time, \(\pi\) units.
3Step 3: Determine the Direction of Motion
The original parametric equations \(x = \sin t\) and \(y = \cos t\) correspond to a counterclockwise motion around the circle. The parameter \(t\) increases, tracing the circle in the counterclockwise direction as \(\sin t\) and \(\cos t\) oscillate between -1 and 1.
4Step 4: Parametric Equations for Opposite Direction
If the particle moves in the opposite direction, or clockwise, we modify the time parameter to \(-t\). Now, the equations become \(x = \sin(-t) = -\sin t\) and \(y = \cos(-t) = \cos t\). These describe clockwise motion around the circle.
Key Concepts
Period of MotionDirection of MotionAngular Speed
Period of Motion
Parametric equations provide crucial insights into the time a particle takes to complete a full journey around a path. Here, the period of motion refers to the time required for the particle to travel once around the circle described by the equations \(x = \sin t\) and \(y = \cos t\). This path is a circle, and both sine and cosine functions inherently have a period of \(2\pi\).
Thus, it takes \(2\pi\) units of time for the particle to complete one trip around the circle. If we wish to increase the speed of the particle, such as making it travel twice as fast, we adjust the parametric equations accordingly. By replacing \(t\) with \(2t\), the equations represent a motion with a reduced period of \(\pi\), leading to a complete circle in half the usual time.
Thus, it takes \(2\pi\) units of time for the particle to complete one trip around the circle. If we wish to increase the speed of the particle, such as making it travel twice as fast, we adjust the parametric equations accordingly. By replacing \(t\) with \(2t\), the equations represent a motion with a reduced period of \(\pi\), leading to a complete circle in half the usual time.
Direction of Motion
Understanding the direction of motion entails examining how the particle traces the path over time. With the original equations \(x = \sin t\) and \(y = \cos t\), as \(t\) increases from 0 to \(2\pi\), the particle moves counterclockwise around the circle.
This counterclockwise, or positive direction, is typical in many parametric equations involving trigonometric functions. However, if you want the particle to move in the opposite, or clockwise, direction, we modify the time parameter by substituting \(t\) with \(-t\).
For instance, \(x = \sin(-t) = -\sin t\) and \(y = \cos(-t) = \cos t\) are the parametric equations that depict a clockwise motion.
This counterclockwise, or positive direction, is typical in many parametric equations involving trigonometric functions. However, if you want the particle to move in the opposite, or clockwise, direction, we modify the time parameter by substituting \(t\) with \(-t\).
For instance, \(x = \sin(-t) = -\sin t\) and \(y = \cos(-t) = \cos t\) are the parametric equations that depict a clockwise motion.
Angular Speed
Angular speed in the context of parametric equations reflects how swiftly a particle traverses its path. It is directly influenced by changes in the parametric variables. In the original equations \(x = \sin t\) and \(y = \cos t\), the particle moves with a constant angular speed, completing the circle in \(2\pi\) time units.
To increase the angular speed, thereby making the particle complete the circle more quickly, we can modify the parameter as \(t\) is replaced by \(2t\). This simple substitution doubles the angular speed, allowing the particle to complete a circle in only \(\pi\) time units.
To increase the angular speed, thereby making the particle complete the circle more quickly, we can modify the parameter as \(t\) is replaced by \(2t\). This simple substitution doubles the angular speed, allowing the particle to complete a circle in only \(\pi\) time units.
- Normal angular speed: Completing the circle in \(2\pi\) time units.
- Increased angular speed (double): Completing the circle in \(\pi\) time units after substitution by \(2t\).
Other exercises in this chapter
Problem 67
Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=-20, \qua
View solution Problem 68
Convert the polar equation to rectangular coordinates. $$ \cos 2 \theta=1 $$
View solution Problem 68
Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=3+4 i, \q
View solution Problem 69
The Distance Formula in Polar Coordinates (a) Use the Law of Cosines to prove that the distance between the polar points \(\left(r_{1}, \theta_{1}\right)\) and
View solution