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Suppose that the position of one particle at time $ t $ is given by$$ x_1 = 3 \sin t \quad y_1 = 1 + \sin t \quad 0 \leqslant t \leqslant 2 \pi $$and the position of a second particle is given by$$ x_2 = 3 + \cos t \quad y_2 = 1 + \sin t \quad 0 \leqslant t \leqslant 2\pi $$(a) Graph the paths of both particles. How many points of intersection are there?(b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points.(c) Describe what happens if the path of the second particle given by$$ x_2 = 3 + \cos t \quad y_2 = 1 + \sin t \quad 0 \leqslant t \leqslant 2\pi $$

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Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

Missouri State University

Oregon State University

Harvey Mudd College

Boston College

Lectures

16:57

In mathematics, a graph is a representation of a set of objects where some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called "vertices" or "nodes", and the relations between them are represented by mathematical abstractions called "edges" or "arcs". The basic notion of a graph was developed by the 17th-century French mathematician Pierre de Fermat, and the term "graph" was coined by the 19th-century mathematician James Joseph Sylvester. The more general mathematical concept of a graph "in which any kind of relation between elements of the set is expressed as an edge, is called a network" (Kolmogorov, "1956, p. 111"). In other words, an undirected graph is a graph in which the edges have no direction associated with them. The most familiar examples of graphs are the graphs of equations. In general, the vertices of a graph can represent concepts and the edges can represent real-valued functions on the concepts, so one can speak of the graph as a function's graph or of the edge as a function's edge.

01:59

Polar coordinates are a two-dimensional coordinate system that specifies a point in terms of distance from a reference direction (the pole) and angle from a reference direction (the polar axis).

01:09

Suppose that the position …

01:54

01:06

03:21

0:00

00:42

Writing$$\begin{array}…

08:45

The paths of two particles…

The problem is suppose that the petitioner for one particle attempt t s given by X one is it goes three times, Scientist E why one is the good one past scientist e He's between zero and two pi on the position of second particle It's given my axe to is you go to negative three plastic society Why to is he got a one class scientist? He is between zero and two high had a graph the path of both particles How many points of intersection are there? So look, cats graph here we have shoo intersection points how to be Are any of this points of intersection relation points if so fine that the collision points So actually both of them are collision points We can prove it as follows So first we can write this parametric equations as why I could too One class acts over three The second one is X plus three squire plus why minus one square is equal to one No and then south This to equations applying here So we have ax plus three squire plus X squared over nine is equal to want self by thousand This equations were half well exploiter us when you're seven. X last thirty six is equal to zero, so Axe is a connective three or X is equal to twelve over negative. Tough over five one acts is a connective. Three. Half he is the cultural three. Pie over to the intersection. Point is negative. Three and zero. If Axe is, they call it connective. Twelve over father, but half sign T. It's the Goto negative war over love and assign T. It's the Call Tio three over five. In this case, the intersection point is negative. Twelve over Fath on DH one over fall. The post is Amar collision points. But see describe what happens if the path of the second a particle I'll even buy three plastic society one class society The first two less locators of graphs. If we change negative three to three, we can see that they are surely in the second points on DH similar to part A and B behalf. Both those, um, are clearly points on DH intersection points are off your father nine over five and three two

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