# Study Notes: Introduction to Parametric Equations Parametric equations are a powerful mathematical tool that allow us to describe the motion and position of objects in terms of a third variable, often time ($t$). This approach is particularly useful in physics, engineering, and other fields where motion or paths need to be analyzed. --- ## What Are Parametric Equations? ### Definition Parametric equations define $x$ and $y$ (or other variables) as functions of a third parameter, typically denoted as $t$. Instead of expressing $y$ directly in terms of $x$ (or vice versa), we express both $x$ and $y$ in terms of $t$. - **General Form**: \[ x = f(t), \quad y = g(t) \] Here, $t$ is the parameter. ### Key Features - **Path Representation**: Parametric equations describe the path of an object in a plane. - **Direction**: They provide information about the direction of motion as $t$ increases. - **Position at Any Time**: By substituting a specific value of $t$, we can determine the exact position of the object. Think of $t$ as a "time stamp" that tells you where the object is at any given moment. --- ## Example: A Car Driving Off a Cliff To understand parametric equations, let’s analyze a car driving off a 50-meter-high cliff at a horizontal speed of 5 m/s. The goal is to determine the car's path and position at any time $t$. ### Setting Up the Problem 1. **Initial Conditions**: - The car starts at $(x, y) = (10, 50)$. - Horizontal velocity: $v_x = 5 \, \text{m/s}$. - No initial vertical velocity ($v_y = 0$). - Acceleration due to gravity: $g = 10 \, \text{m/s}^2$ (approximated for simplicity). 2. **Coordinate System**: - $x$-axis: Horizontal position. - $y$-axis: Vertical position (height). --- ### Deriving the Parametric Equations #### Horizontal Motion ($x$): - The car moves horizontally at a constant speed of $5 \, \text{m/s}$. - Using the formula for distance, $x = v_x t + x_0$, where $x_0$ is the initial position: \[ x(t) = 5t + 10 \] #### Vertical Motion ($y$): - The car is affected by gravity, which accelerates it downward. - Using the kinematic equation for vertical motion: \[ y(t) = y_0 + v_y t - \frac{1}{2} g t^2 \] Substituting $y_0 = 50$, $v_y = 0$, and $g = 10 \, \text{m/s}^2$: \[ y(t) = 50 - 5t^2 \] --- ### Parametric Equations for the Car The motion of the car is described by: \[ x(t) = 5t + 10, \quad y(t) = 50 - 5t^2 \] --- ## Analyzing the Motion ### Table of Values Let’s calculate the car’s position at different times: | $t$ (seconds) | $x(t)$ (meters) | $y(t)$ (meters) | |---------------|-----------------|-----------------| | 0 | $10$ | $50$ | | 1 | $15$ | $45$ | | 2 | $20$ | $30$ | | 3 | $25$ | $5$ | At $t = 2$ seconds: - $x(2) = 5(2) + 10 = 20$ - $y(2) = 50 - 5(2)^2 = 50 - 20 = 30$ Thus, the car is at $(20, 30)$. --- ### Graphing the Path 1. Plot the points $(x, y)$ for each value of $t$. 2. Connect the points to form a smooth curve. 3. Add arrows to indicate the direction of motion as $t$ increases. The resulting path is a **parabolic curve** opening downward, representing the car’s trajectory. --- ### Key Observations - The horizontal motion is linear because there is no horizontal acceleration. - The vertical motion is quadratic due to the constant acceleration of gravity. - The car’s path is a combination of these motions, forming a parabola. Parametric equations allow us to describe the motion in two dimensions without directly relating $x$ and $y$. --- ## Eliminating the Parameter To express $y$ directly in terms of $x$, eliminate $t$ from the parametric equations: 1. Solve for $t$ in the $x$ equation: \[ t = \frac{x - 10}{5} \] 2. Substitute into the $y$ equation: \[ y = 50 - 5\left(\frac{x - 10}{5}\right)^2 \] 3. Simplify: \[ y = 50 - \frac{(x - 10)^2}{5} \] This is the equation of the parabola describing the car’s path. Eliminating the parameter gives the overall shape of the path, but it loses information about direction and time. --- ## Why Use Parametric Equations? 1. **Direction and Time**: Parametric equations show the direction of motion and allow us to calculate the position at any specific time. 2. **Complex Motion**: They simplify problems involving motion in multiple dimensions. 3. **Real-World Applications**: Used in physics (projectile motion), engineering (robotics), and computer graphics (animation). In 3D motion, parametric equations can describe $x$, $y$, and $z$ as functions of $t$: \[ x(t) = f(t), \quad y(t) = g(t), \quad z(t) = h(t) \] This is essential for modeling trajectories in space. --- ## Common Misconceptions **Confusing $t$ with $x$ or $y$**: Remember, $t$ is an independent parameter, not a spatial coordinate. **Forgetting the direction**: Without the parameter, you lose information about the direction of motion. **Assuming $t$ must represent time**: While $t$ often represents time, it can represent other quantities like angle or distance. --- ## Practice Problems 1. A ball is thrown horizontally from a height of 20 meters with a speed of 10 m/s. Write the parametric equations for its motion and determine its position at $t = 2$ seconds. 2. Eliminate the parameter from the following equations: \[ x(t) = 3t + 2, \quad y(t) = 4t^2 - 1 \] 3. Sketch the path of an object described by: \[ x(t) = 2t, \quad y(t) = 10 - t^2 \] --- ## Key Takeaways - Parametric equations describe motion in terms of a third parameter, often time. - They provide a complete picture of an object’s path, direction, and position. - Eliminating the parameter gives the overall shape of the path but loses time-related details. - They are widely used in physics, engineering, and beyond to model real-world scenarios. Parametric equations are your go-to tool for analyzing motion in multiple dimensions!