Learning objective
A parametric function consists of a pair of functions, given by f(t) = (x(t),\, y(t)), where x(t) and y(t) are functions of an independent variable t, also known as the parameter. In the context of particle motion, the parameter t often represents time, while the dependent variables x and y indicate the particle's position in the plane.
The graph of a parametric function traces the path of the particle as it moves through the plane over time. Each point on the graph corresponds to a specific position of the particle at a particular time, represented by the coordinates (x(t),\, y(t)).
To analyze the motion of the particle, we can examine the individual parametric equations x(t) and y(t). These equations describe the horizontal and vertical components of the particle's motion, respectively. By studying these equations, we gain valuable insights into the behavior of the particle and its trajectory in the plane. The maximum and minimum values of the functions x(t) and y(t) correspond to the horizontal and vertical extrema of the particle's motion, respectively. These extrema represent the farthest points the particle reaches in the horizontal and vertical directions. By identifying these extrema, we can better understand the overall shape and characteristics of the particle's path.
A particle moves in the plane following the given parametric equations: x(t) = 4t - 3 and y(t) = t^{2} + 1, for t in the domain [0,\, 3].
Find the parametric function f(t) that represents the motion of the particle.
A particle is moving along a trajectory described by the following parametric equations: \\x(t) = 3\cos(t) and y(t) = 4\sin(t).
Determine the position of the particle at t = \dfrac{\pi}{2}.
Graph the path of the particle for t in the interval [0,\, 2\pi].
A parametric function consists of a pair of functions, given by f(t) = (x(t),\, y(t)), where x(t) and y(t) are functions of an independent variable t, also known as the parameter.
We will explore how to identify the horizontal and vertical extrema of a particle's motion using the functions x(t) and y(t). Extrema are the maximum and minimum values that represent the furthest points reached by the particle in the horizontal and vertical directions.
To determine the horizontal extrema, we focus on the function x(t) which represents the horizontal position of the particle at time t. The maximum value of x(t) corresponds to the furthest point reached by the particle in the positive x-direction, while the minimum value represents the furthest point in the negative x-direction.
Similarly, to find the vertical extrema, we examine the function y(t) which represents the vertical position of the particle. The maximum value of y(t) indicates the highest point reached by the particle, while the minimum value represents the lowest point.
Alternatively, for certain types of functions, we can use graphical analysis to identify the extrema. Graphical analysis is a method used to visually identify the extrema of a particle's motion in the plane by plotting the parametric equations x(t) and y(t) on a coordinate system.
Consider a particle moving in the xy-plane, with its position at time t given by the functions \\x(t) = t^{2} - 2t and y(t) = t + 1. Find the horizontal and vertical extrema of the particle's motion.
A particle moves in the xy-plane with its position given by the functions x(t) = 2 \sin(t) and \\y(t) = \cos(t) for t in the interval [0,\, 2\pi]. Find the horizontal and vertical extrema of the particle's motion.
These maximum and minimum values correspond to the horizontal and vertical extrema of the particle's motion, representing the farthest points the particle reaches in the horizontal and vertical directions. By identifying these extrema, we can better understand the overall shape and characteristics of the particle's path.
In this section, we focus on finding intercepts of parametric functions modeling particle motion in the plane. The x-intercept of a parametric function occurs when the particle's path crosses the x-axis (y(t) = 0), and the y-intercept occurs when the path crosses the y-axis (x(t) = 0). These intercepts provide important information about the particle's motion and can help us better understand its trajectory.
Given the parametric equations x(t) = t^{2} - 4t and y(t) = 2t - 4, find the x-intercept and y-intercept of the particle's path.
Find the x-intercept and y-intercept of the particle's path given by the parametric equations \\x(t) = 3t - 1 and y(t) = t^{3} - t.
To find the x-intercepts, we set the y(t) function equal to zero and solve for the parameter t. We substitute the values of t back into the x(t) function to find the corresponding x-values.
To find the y-intercepts, we set the x(t) function equal to zero and solve for t. Then, we substitute these t-values into the y(t) function to find the corresponding y-values.
By finding the intercepts, we gain a better understanding of the particle's trajectory and how it interacts with the coordinate axes.