Question

(a) Each set of parametric equations represents the motion of a particle. Use a graphing utility to graph each set. $$ \frac{\text { First Particle }}{x=3 \cos t} \quad \frac{\text { Second Particle }}{x=4 \sin t} $$ \(y=4 \sin t \quad y=3 \cos t\) $$ \begin{array}{ll} 0 \leq t \leq 2 \pi & 0 \leq t \leq 2 \pi \end{array} $$ (b) Determine the number of points of intersection. (c) Will the particles ever be at the same place at the same time? If so, identify the points. (d) Explain what happens if the motion of the second particle is represented by \(x=2+3 \sin t, \quad y=2-4 \cos t, \quad 0 \leq t \leq 2 \pi\)

Short answer

This answer will depend on the specific solutions to the equations presented in each step. The intersection and coincidence points specifically will be determined by solving the equations. The analysis of the change in the second particle's path will also be specific to the outcomes of the new equation for this path.

Step-by-step solution

Graph the Parametric Equations

Use your graphing calculator or corresponding software on your computer to plot the two sets of parametric equations. The first particle has equations \(x=3 \cos t\) and \(y=4 \sin t\), and the second one has \(x=4 \sin t\) and \(y=3 \cos t\). For both particles, the parameter \(t\) ranges from 0 to \(2\pi\). By plotting these, you can visualize the paths of the two particles.

Determine Points of Intersection

To find the points where the paths of the two particles intersect, you need to set the x-coordinates equal to each other and similarly the y-coordinates. So you need to solve the system: \(3\cos t = 4\sin t\) (for \(x\)) and \(4\sin t = 3\cos t\) (for \(y\)). Solve these equations for \(t\), and substitute these values of \(t\) back into the original parametric equations to find the points of intersection.

Check for Coincidence

To determine if the particles ever occupy the same place at the same time, you need to solve simultaneously the parametric equations of both particles for the same value of \(t\). If there is a value of \(t\) that satisfies all four equations, then there is a point in time where the two particles would be at the same place.

Analyse Second Particle's Motion

The motion of the second particle is altered by the set of equations: \(x=2+3 \sin t\), \(y=2-4 \cos t\) and \(0\leq t\leq 2\pi\). Graph these equations and compare it with the initial path. Analyze whether the points of intersection or coincidence points with the first particle's path change.

Key concepts

Graphing Parametric Equations

Parametric equations are a way to express a mathematical relationship by using parameters. In the context of graphing, these equations define both the x and the y coordinates as functions of a third variable, often time (t). To graph parametric equations like
\(x = 3 \cos t\) and \(y = 4 \sin t\),
we treat the variable \(t\) as the input and the x and y coordinates as the outputs. As \(t\) changes within its specified range, which is usually a segment of real numbers, it traces out a path on the coordinate plane representing the motion of a particle.

Graphing parametric equations involves evaluating the x and y equations separately over a range of values of the parameter, then plotting the resulting coordinate pairs. Software or a graphing calculator can facilitate this process, plotting a smooth curve that shows the trajectory of the particle represented by the equations. It's crucial to consider the range of \(t\), as it dictates the portion of the curve that is relevant to the problem.

Points of Intersection

Points of intersection are the points where two or more graphed lines or curves meet. To find these points between two parametric equations, we look for parameter values that lead to the same x and y coordinates in both sets of equations. This generally involves solving a system of equations
\(3\cos t = 4\sin t\) and \(4\sin t = 3\cos t\),
which are derived from equating the parametric expressions for x and y from each particle's path. By finding the values of \(t\) that satisfy both equations, you identify the parameter values at which the intersection occurs. Substituting these \(t\) values back into the original parametric equations yields the exact coordinates of the points of intersection. Understanding these intersections is critical, as they often represent significant events or solutions to applied problems in various fields such as physics, engineering, and computer graphics.

Particle Motion in Calculus

Particle motion in calculus refers to using equations to describe the movement of a particle through space over time. When using parametric equations to model particle motion, each value of the parameter \(t\), which often represents time, generates a point \((x(t), y(t))\) that corresponds to the particle's position on the plane at that instant.

This approach is powerful for modeling complex movements, such as those involving curves that change direction and speed over time. Calculus tools like differentiation and integration are applied to parametric equations to analyze properties of motion like velocity, acceleration, and the length of the path traveled. For instance, differentiating the position functions with respect to time provides velocity, and further differentiation yields acceleration. These concepts are foundational across physical sciences and engineering, where the movement of objects needs to be precisely controlled or predicted.

Solving Parametric System

Solving a parametric system involves finding a set of parameter values that satisfy multiple parametric equations simultaneously. In the context of our exercise, we want to see if the point where the two particles described by separate parametric equations are at the same place at the same time. This requires setting up a system where the x and y coordinates defined by the equations are equal.

For two particles with parametric equations \(x_1(t), y_1(t)\) and \(x_2(t), y_2(t)\), if we can find a value of \(t\) such that \(x_1(t) = x_2(t)\) and \(y_1(t) = y_2(t)\), then we have a solution to the system, and the particles intersect. In the given problem, we analyze the scenarios before and after changing the second particle's motion. Solving such a system illustrates the changes in dynamics of particle motion and helps to understand the behavior of moving objects in both theoretical and real-world situations.