Name: modeling baseball motion with parametric curves
Uploaded: 2026-06-08T09:54:19.187Z
Duration: 1 min 21 s
Channel: JoVE

When a baseball is hit, ignoring air resistance, its path through the air can be shown by a downward-facing parabola.Here, x is the horizontal position along this path, and y is the height above the ground.To describe this movement accurately, the ball&#x2019;s position is expressed using a third variable: time t.Instead of a single equation like y = f(x), x and y are each written as separate functions of time, x = f(t) and y = g(t). These are known as parametric equations.This approach is essential because a spatial function like y = f(x) only shows the curve, not the ball&#x27;s progress along it. Parametric equations allow an observer to track the ball&#x27;s exact location at any specific time.This becomes particularly useful when the ball reaches the same height at two different times&#x2014;once while rising at t1 and again while falling at t2. By defining each point as the ordered pair (f(t), g(t)), the parameter t traces the entire trajectory and simultaneously models the motion.

curves defined by parametric equations

A baseball hit into the air follows a parabolic trajectory when air resistance is neglected. The motion can be described within a two-dimensional coordinate system, where both the horizontal displacement and vertical height are functions of time. Instead of expressing the trajectory as a single function of position, the motion is modeled using parametric equations: one function for the horizontal position and another for the vertical position as time progresses. Let the horizontal position be defined as f(t) and the vertical position as g(t); these functions represent the x- and y-coordinates of the baseball at any time t.The use of parametric equations allows a complete description of the baseball&#x27;s trajectory. The function f(t) gives the horizontal position based on the constant horizontal velocity, while g(t) incorporates both the initial vertical velocity and the downward acceleration due to gravity. This representation is essential because it accurately accounts for the baseball&#x27;s position at each moment, resolving the ambiguity present in a single height corresponding to two different times&#x2014;once during ascent and again during descent.In the context of space alone, the baseball&#x2019;s path appears parabolic; however, when represented parametrically, each point on the trajectory corresponds to a unique time value. So, the pair (f(t), g(t)) provides a one-to-one mapping between time and position in two-dimensional space. Although this method does not yield a single function y = f(x), it ensures a more complete and time-specific analysis of the motion, which is crucial for applications requiring precise tracking of position over time.

Curves Defined by Parametric Equations

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Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Parametric Equations and Polar Coordinates

المعادلات البارامترية والإحداثيات القطبية

parametric-equations-and-polar-coordinates

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

المنحنيات المعرفة بالمعادلات البرامترية

modeling baseball motion with parametric curves

Baseball motion can be modeled with parametric equations when air resistance is neglected. The path looks parabolic in a two-dimensional coordinate system, but the full motion is described as position changing with time.

In this model, the horizontal position is written as f(t) and the vertical position as g(t). These functions give the x- and y-coordinates of the baseball at any time t. The horizontal part reflects constant horizontal velocity. The vertical part includes the initial vertical velocity and the downward acceleration caused by gravity.

Parametric equations are useful because they show the baseball’s location at each moment. They also remove the ambiguity that comes from a single height value appearing at two different times, once on the way up and once on the way down. In space alone, the path appears as a parabola.

When the curve is written parametrically, each point on the trajectory matches one unique time value. The pair (f(t), g(t)) gives a one-to-one link between time and position in two-dimensional space. This method does not produce a single equation of the form y = f(x), but it gives a more complete, time-specific description of the motion. That makes it important for precise tracking of position over time.

Modeling Baseball Motion with Parametric Curves

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key terms and concepts

See how a baseball’s path is modeled with parametric equations, using time to track horizontal and vertical motion in two dimensions.

Modeling Baseball Motion with Parametric Curves | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

Modeling Baseball Motion with Parametric Curves

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