Problem 68 - Precalculus Mathematics for Calculus

Question

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

Answer

Initial time is \(2\pi\); twice speed: \( x=\sin(2t), y=\cos(2t) \); counterclockwise.

Step 1: Find the Initial Time Period

The given parametric equations are \( x = \sin t \) and \( y = \cos t \). These equations describe a circle because as \( t \) varies from 0 to \( 2\pi \), \( x^2 + y^2 = \sin^2 t + \cos^2 t = 1 \), which is the equation of a circle.For one complete revolution, the particle takes \( t \) from 0 to \( 2\pi \). Thus, the initial time period for one full circle is \( 2\pi \) units of time.

Step 2: Find New Parametric Equations for Double Speed

To double the speed, we need to make one revolution in half the time, or \( \pi \) units of time.We achieve this by adjusting \( t \) in the parametric equations to \( 2t \). This is because the angle completed would be \( 2t \) when \( t \) varies from 0 to \( \pi \), thus covering \( 0 \) to \( 2\pi \).The new parametric equations become:\[ x = \sin(2t) \]\[ y = \cos(2t) \]

Step 3: Determine the Direction of Travel

The initial parametric equations \( x = \sin t \) and \( y = \cos t \) imply a counterclockwise motion. This is because when \( t = 0 \), the point is at \((0, 1)\), and as \( t \) increases, the direction follows counterclockwise.

Step 4: Find Parametric Equations for the Opposite Direction

To reverse the direction, the particle must move clockwise. One way to reverse the direction is to simply negate the parameter \( t \) in one of the trigonometric functions.The new parametric equations for clockwise motion become:\[ x = \sin(-t) = -\sin t \]\[ y = \cos(-t) = \cos t \]Another equivalent solution would be to switch the functions:\[ x = \cos t \]\[ y = \sin t \]

Key Concepts

Particle Motion

Particle motion refers to the continuous movement of a particle along its path, which can be described using mathematical equations. In this context, parametric equations are often used to represent the position of a particle over time. These equations use a variable, often representing time like in our example where \( t \) represents time, to define both the \( x \) and \( y \) coordinates of the particle position.

Understanding particle motion using parametric equations provides more than just the path's shape. It also gives us insight into how the particle moves along that path. For instance, with the circular motion described by \( x = \sin t \) and \( y = \cos t \), we can determine not only that the path is a circle but also the direction (counterclockwise) and the speed at which the particle travels the circle, based on how \( t \) changes over time.

• The direction of motion shows how the particle travels around the path, clockwise or counterclockwise.
• The speed can be adjusted by modifying the rate of change of \( t \), affecting the length of time required for one complete revolution.

Circular Motion

Circular motion occurs when an object moves in a circular path. It's a common phenomenon, seen in a variety of systems, and it can be effectively described using parametric equations. In the given exercise, the particle follows a circular path with its position expressed as \( x=\sin t \) and \( y=\cos t \), representing a circle with a radius of 1.

The parametric form allows us not only to see the shape of the motion path—a circle—but also to calculate how long a particle takes to complete a full rotation. Initially, the particle takes from \( t = 0 \) to \( 2\pi \) to make one full circle, implying counterclockwise circular motion. Adjusting the speed, by changing the parameter to \( 2t \), results in the particle moving twice as fast; it completes the circle in half the time, demonstrating how parametric equations can be manipulated to alter motion characteristics.

• Counterclockwise circular motion is indicated by the increase of \( t \) starting from \((0,1)\).
• We can reverse circular direction by changing the parameters, for example, switching \( x \) and \( y \).

Time Parameterization

Time parameterization involves using a variable, often time, to describe the motion of a particle along a path. In our example, the parameter \( t \) is used in the equations \( x = \sin t \) and \( y = \cos t \) to trace the particle's circular trajectory.

This time parameter helps in multiple ways:

• It defines the speed of the particle's motion along the path. Faster speeds can be achieved by modifying \( t \), for instance, using \( 2t \) for double speed.
• Direction is determined by the progression of \( t \). A positive increase in \( t \) indicates the motion's direction, either clockwise or counterclockwise.

The power of parametric equations and time parameterization lies in their ability to fully describe a particle's dynamic behavior, from speed adjustments to direction alterations, all while maintaining a clear representation of the path taken.