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

当击打棒球时，若忽略空气阻力，其在空中的运动轨迹可用一个开口向下的抛物线表示。此处，x 是沿该路径的水平位置，y 是距地面的高度。为了准确描述这一运动，球的位置需使用第三个变量：时间 t。与像 y = f(x) 这样的单一方程不同，x 和 y 分别被写成关于时间的独立函数，即 x = f(t) 和 y = g(t)。这类方程称为参数方程。这种方法至关重要，因为像 y = f(x) 这样的空间函数仅能显示曲线的形状，而无法反映球体沿曲线运动的过程。参数方程则允许观察者追踪球体在任意特定时刻的精确位置。当球在两个不同的时刻达到相同高度时——一次是在上升过程中（t1），另一次是在下落过程中（t2）——这一方法尤为有用。通过将每个点定义为有序对（f(t)，g(t)），参数 t 可追踪整个运动轨迹，同时对运动过程进行建模。

由参数方程定义的曲线

Curves Defined by Parametric Equations

当忽略空气阻力时，击向空中的棒球遵循抛物线轨迹。该运动可以用二维坐标系来描述，其中水平位移和垂直高度均为时间的函数。轨迹并非表示为单一的位置函数，而是通过参数方程建模：一个函数描述水平位置，另一个函数描述随着时间推移的垂直位置。设水平位置为 f(t)，垂直位置为 g(t)；这些函数表示棒球在任意时刻 t 的 x 和 y 坐标。使用参数方程可以完整描述棒球的运动轨迹。f(t) 函数根据恒定的水平速度给出水平位置，而 g(t) 函数则同时包含初始垂直速度以及向下的重力加速度。这种表示方法至关重要，因为它能够准确描述棒球在每一时刻的位置，从而解决了同一高度对应两个不同时间点（一次在上升阶段，另一次在下降阶段）所导致的歧义问题。仅从空间角度来看，棒球的运动轨迹呈现抛物线形状；然而，当采用参数化表示时，轨迹上的每一点都对应一个唯一的时间值。因此，数对 (f(t), g(t)) 实现了时间与二维空间中位置之间的一一映射。尽管这种方法无法得到单一的函数关系 y = f(x)，但它能够对运动进行更完整且与时间精确对应的分析，这对于需要精确追踪位置随时间变化的应用至关重要。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Parametric Equations and Polar Coordinates

参数方程与极坐标

parametric-equations-and-polar-coordinates

KEY TERMS AND CONCEPTS

关键术语与概念

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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curves defined by parametric equations

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...

calculus with parametric curves: tangents and areas

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Parametric Curves: Slope and Tangent Lines

Parametric curves use two functions, x(t) and y(t), to show the position of a point in the plane. The parameter t often stands for time. This setup helps describe both the path of the point and the direction it moves. The slope of a tangent line tells how steep the curve is at one point. For a parametric curve, the slope is found with dy/dx. Because x and y both depend on t, the chain rule gives the formula dy/dx = (dy/dt) / (dx/dt), as long as dx/dt is not zero. The derivatives also reveal...

calculus with parametric curves: surface areas

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Parametric Curves and Rotated Surfaces

Parametric curves can be used to find the surface area of a shape made by rotation. A parametric curve describes a path in the plane with both x and y as functions of one parameter, usually t. When that curve is revolved around an external axis in the same plane, it creates a surface of revolution in three dimensions. The surface area depends on the shape of the curve and on how far the curve sits from the axis of rotation. A torus is a classic example of this kind of surface. It forms when a...

polar coordinate system

Mapping Points with Polar Coordinates

Polar coordinates map points in a plane by using distance and direction from a fixed center. This system is useful when a shape has circular or spiral symmetry, because those forms are easier to describe than they are on a rectangular grid. It can also help with regions that are not naturally rectangular. A ship’s plan position indicator is a familiar example. The ship sits at the center of the display, and detected targets appear as dots around it. In polar form, that center point is called...

polar curves

Spirograph Patterns in Polar Coordinates

Spirograph patterns show how polar coordinates can describe curves with angles and distances. Polar coordinates use a radius r, which is the distance from the origin, and an angle θ, measured counterclockwise from the polar axis. This system gives a useful alternative to Cartesian coordinates, which use x and y values. A polar curve is written as an equation in the form r = f(θ). As θ changes, the radius changes too, so each point moves farther from or closer to the origin. This makes polar...

integration applied to polar coordinates to find areas

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Polar Area of an Irregular Sprinkler Spray

Polar coordinates can be used to measure the area covered by an irregular sprinkler spray. In this setup, the spray distance changes with direction, so the radius is treated as a function of the angle of rotation. That changing radius creates a polar curve, which maps the sprinkler’s uneven reach across a full turn. To find the watered region, the area is split into many thin sectors. A sector is a narrow slice of a circle. Each slice has an area that depends on both its radius and the angle...

integration applied to polar coordinates to find arc lengths

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Polar Curve Distance with Calculus

Polar coordinates can be used to measure the distance along a curve that turns around a fixed point. In this system, a point is described by the radial distance r from the pole and the angle θ from a reference direction. That makes it useful for motion such as an expanding spiral, where distance and direction change together. A search drone gives a clear example of a polar path. If the hiker’s last known position is the pole, the drone’s location at any moment can be written as the polar...

polar coordinates: problem solving

Directional Beam Area in Polar Plots

Directional radiation patterns use polar plots to show how signal strength changes with direction. In a polar plot, the radial distance from the origin represents signal intensity at a given angle. This makes the graph useful for antenna analysis and for studying focused beams. A simple model for this kind of pattern is the four-lobed rose curve. It is described by r = cos(2θ) and has four symmetric lobes. Each lobe acts like a focused transmission beam in a simplified mathematical form. To...

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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