4.2 Parametric Functions Modeling Planar Motion

Parametric functions can be used to model planar motion (bodies or objects that are both rotating and translating at the same time), in which we can analyze the position, velocity, and acceleration of an object in two dimensions. 🚗

These functions can be used to represent the motion of an object in terms of its x and y coordinates as a function of time.

For example, if an object is moving in a circular path, the x and y coordinates can be represented as the sine and cosine of the angle of the object's position around the circle. Additionally, the velocity and acceleration of the object can be determined by taking the derivatives of the x and y coordinates with respect to time. 💫

This information can then be used to analyze the motion of the object, such as determining the speed and direction of the object at a particular point in time. Furthermore, these functions can also be used to predict the future position, velocity and acceleration of the object. ✍️

🧑‍🔧 Modeling Time!

The x(t) and y(t) functions represent the coordinates of the particle's position at time t. The domain of t can represent a certain time interval in which the motion occurs. For example, if t is in the domain of [0, 10], this means the particle's motion is being modeled for a time period of 10 seconds, starting from time t = 0. ⏱️

Additionally, the graph of the parametric function can be used to visualize the path of the particle, which can be useful in understanding the motion. 👁 For example, if the x(t) and y(t) functions are both sinusoidal, the particle's motion would be a simple harmonic motion in a circular path.

🔃 Horizontal and Vertical Extrema

The horizontal and vertical extrema of a particle’s motion can be determined by identifying the maximum and minimum values of the functions x(t) and y(t), respectively.

In the case of a particle's motion, the graph of the parametric function represents the path of the particle in the plane. The horizontal and vertical extrema of the motion are the maximum and minimum values of the x-coordinate and y-coordinate, respectively. These extrema represent the farthest points that the particle reaches in the horizontal and vertical directions during its motion. 👐

To find the horizontal extrema, we need to identify the maximum and minimum values of the x(t) function. Similarly, to find the vertical extrema, we need to identify the maximum and minimum values of the y(t) function. 🧐

Source: Math LibreTexts

For example, if the domain of the parametric function is given by 0 < t < 10, we can choose several values of t within this domain and evaluate the x(t) and y(t) functions. By comparing the output of these evaluations, we can determine which value of x(t) is the largest and which is the smallest, and similarly for y(t). These values correspond to the horizontal and vertical extrema of the motion! 🤝

Alternatively, we can use algebraic methods to find the extrema of x(t) and y(t) functions. This is done by analyzing the shape of the function, looking for patterns and identifying any symmetry or periodicity in the function. 🔎

For example, if a function is periodic, it will repeat itself after a certain interval, and the extrema will be reached at the beginning and end of the interval. Similarly, if a function is symmetric, the extrema will be reached at the center of the symmetry. These methods provide a way to identify the extrema of the functions without having to evaluate them for different values of t. 😁

📈 Intercepts

The points at which the function x(t) crosses the x-axis (y = 0) correspond to the y-intercepts, and the points at which the function y(t) crosses the y-axis (x = 0) correspond to the x-intercepts. In other words, the real zeros of x(t) and y(t) correspond to the points at which the particle's motion intercepts the x-axis and y-axis, respectively. 📌

Source: Massey University

The real zeros of x(t) are the values of t for which x(t) = 0, and the real zeros of y(t) are the values of t for which y(t) = 0. These values of t correspond to the values of t at which the particle's motion crosses the x-axis and y-axis, respectively. So, the real zeros of x(t) correspond to the y-intercepts and the real zeros of y(t) correspond to the x-intercepts.

For example, if the parametric function is given by f(t) = (t^2 - 4, t), the x-intercepts would be the points where y = 0. In this case, y(t) = t = 0, so t = 0 is the x-intercept. The y-intercepts would be the points where x = 0, in this case, x(t) = t^2 - 4 = 0. so t = 2, -2 are the y-intercepts! ✅