Problem 45 - A Graphical Approach to College Algebra

Question

Path of a Projectile \(\quad\) A projectile moves so that its position at any time \(t\) is given by the equations $$x=60 t \text { and } y=80 t-16 t^{2}$$ Graph the path of the projectile, and find the equivalent rectangular equation. Use the window \([0,300]\) by \([0,200]\)

Step-by-Step Solution

Answer

The rectangular equation is \( y = \frac{4x}{3} - \frac{x^2}{225} \). Graph this equation from \( x = 0 \) to \( x = 300 \).

Step 1: Convert Time Equation to Rectangular Form

First, express time, \( t \), in terms of \( x \) using the equation \( x = 60t \). Solving for \( t \) gives \( t = \frac{x}{60} \).

Step 2: Substitute Time Equation into Y Equation

Substitute \( t = \frac{x}{60} \) into the equation for \( y \): \( y = 80t - 16t^2 \). This becomes \( y = 80\left(\frac{x}{60}\right) - 16\left(\frac{x}{60}\right)^2 \). Simplify this equation to find the relationship between \( y \) and \( x \) without \( t \).

Step 3: Simplify the Equation

Simplifying \( y = 80\left(\frac{x}{60}\right) - 16\left(\frac{x^2}{3600}\right) \) results in \( y = \frac{4x}{3} - \frac{x^2}{225} \). This is the rectangular equation of the path.

Step 4: Graph the Path of the Projectile

Use the rectangular equation \( y = \frac{4x}{3} - \frac{x^2}{225} \) to graph the path. Set the window of the graph with \( x \) ranging from 0 to 300 and \( y \) from 0 to 200. Plot the graph using a graphing calculator or software.

Key Concepts

Rectangular Equation

In mathematics, a rectangular equation is an expression that represents a relationship between two variables, usually denoted as \( x \) and \( y \). It is a standard form of expressing an equation where each term is explicitly dependent on \( x \) and \( y \) only, without involving another variable such as \( t \).
For projectile motion problems like the one given, converting from parametric forms to rectangular forms allows us to visualize the trajectory using familiar Cartesian coordinates.
Bullet points to keep in mind when working with rectangular equations:

• Identify the dependent and independent variables in the equations.
• Express any dependency (time, parameter) just in terms of \( x \) and \( y \).
• Simplify to reduce fractions and ensure the equation is in the simplest form possible.

In this exercise, by substituting \( t = \frac{x}{60} \) into the equation for \( y \), we eliminate \( t \) and find an equation that relates \( y \) directly to \( x \), giving us \( y = \frac{4x}{3} - \frac{x^2}{225} \). This is the projectile's path in rectangular form.

Graphing Calculator

A graphing calculator is an immensely useful tool for students and professionals dealing with complex equations like those in projectile motion problems. These calculators enable you to visualize the trajectory or shape of equations quickly on a coordinate plane.
This visualization becomes especially pertinent with our exercise, as it allows a clear interpretation of how the projectile's path behaves.
Here are a few key features of graphing calculators:

• They can plot multiple equations simultaneously, allowing comparisons and overlaps.
• Settings enable adjustments to the viewing window for tailored graph scale—relevant here with a setting from 0 to 300 for \( x \) and 0 to 200 for \( y \).
• Advanced models even support symbolic solving, letting them process algebraic solutions directly.

In our projectile motion scenario, inputting the rectangular equation \( y = \frac{4x}{3} - \frac{x^2}{225} \) into the graphing calculator helps visualize the parabola that represents the projectile's path.

Parametric Equations

In the realm of mathematics, parametric equations are a way of expressing a set of related quantities as explicit functions of an independent parameter, often time \( t \). Unlike rectangular equations, which directly relate \( x \) and \( y \), parametric equations allow tracking of each variable over a third common variable.
This is particularly beneficial in physics applications, such as the motion of projectiles, where separate equations can describe horizontal and vertical positions over time.
Consider these points regarding parametric equations:

• They typically form a set (e.g., one for \( x \) and one for \( y \)), both dependent on \( t \).
• Conversion to rectangular form requires solving one equation for \( t \) and substituting it into the second.
• The parameter often represents something practical, like time, enabling more flexible modeling.

In the exercise, parametric equations \( x = 60t \) and \( y = 80t - 16t^2 \) define a projectile's path over time, beautifully illustrating how these equations effectively handle dynamic systems.