Name: vector functions in motion and displacement
Uploaded: 2026-06-08T09:38:47.452Z
Duration: 1 min 23 s
Channel: JoVE

考虑一个在空间中运动的无人机，其速度随时间变化。速度可以表示为向量值函数，其中每个标量分量均为实值函数，用于描述在特定方向上的运动。在二维空间中，无人机的运动可以用速度函数的水平分量和垂直分量来表示。通过对各分量分别进行积分，可得到该函数的不定积分，从而获得一个新的向量函数，该函数表示任意时刻 t 的位置。在定积分中，将积分限应用于每个标量分量，可得到在指定时间区间内沿 x 轴和 y 轴的位置净变化量。在三维空间中，一个额外的分量用于描述沿第三根轴的运动。速度函数的每个分量被独立积分。通过此过程，确定了三维空间中的完整位置函数。当对无人机的矢量速度函数在确定的积分限内进行积分时，所得结果是一个新矢量，表示无人机在三个空间方向上的总位移。

向量函数的积分

向量值函数为描述空间中同时涉及大小和方向的运动提供了便捷的框架。无人机在任意时刻的速度都具有方向和速率，而随着无人机的移动，这两个量都可能发生变化。向量值函数通过为每个时间点分配一个向量来刻画这种行为，该向量的各个分量均为实值函数。每个分量表示沿空间中某一特定坐标轴的运动。根据向量中包含的分量数量，这类函数可用于描述二维或三维空间中的运动。在二维情况下，无人机的速度由两个分量函数描述：一个表示水平方向的运动，另一个表示竖直方向的运动。为了求出无人机在任意时刻的位置，需对速度函数关于时间进行积分。该积分通过分别对每个分量函数进行。结果是一个新的矢量函数，其分量给出了无人机在时刻 t 的水平和竖直位置。积分常数表现为一个常矢量，对应于参考时刻无人机的初始位置。不定积分表示与给定速度函数相关联的一组可能的位置函数。从物理意义上讲，它描述了物体在任意时刻的位置，同时包含了其初始位置的信息。当需要计算物体在一段时间间隔内的位置净变化时，需使用定积分。通过对每个分量函数应用积分限，可得到该时间间隔内水平和垂直方向位置的总变化量。这些数值构成一个向量，表示无人机在平面上的净位移。位移向量同时指示了位置总体变化的方向和距离。定积分表示物体在指定时间间隔内的总位移。从物理意义上讲，它度量的是初始时刻与最终时刻之间的整体位置变化，而非某一瞬时的位置。在三维空间中，运动包含一个对应于第三轴的额外分量。因此，速度函数具有三个分量，分别表示沿每个空间方向的运动。与二维情况类似，每个分量均独立进行积分。所得的位置函数综合了全部三个位置分量，描述了无人机在整个空间中的运动。当在特定时间区间内进行计算时，该积分产生一个位移矢量，表示无人机在三维空间中的总位置变化。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Vector Functions and Motion

向量函数与运动

KEY TERMS AND CONCEPTS

关键术语与概念

Integrals of Vector Functions

vector functions in motion and displacement

Vector functions describe motion in space when both speed and direction matter. A drone is a clear example, because its velocity changes as it moves. A vector-valued function assigns a vector to each time value. Each component is a real-valued function that tracks motion along one axis in space.

These functions can model motion in two dimensions or three dimensions. In two dimensions, the velocity of the drone has two component functions. One component shows horizontal motion, and the other shows vertical motion. In three dimensions, there is an additional component for the third axis, so the motion can be described in full space.

To find position from velocity, the velocity function is integrated with respect to time. Each component is integrated separately. The result is a new vector function whose components give the position along each axis at time t. The constant of integration appears as a constant vector. This constant vector represents the drone’s initial position at the reference time.

The indefinite integral gives the family of possible position functions for a given velocity function. In physical terms, it shows the object’s position at any instant together with its starting point. When the net change in position over a time interval is needed, definite integration is used instead. The limits of integration are applied to each component, and the result is a displacement vector in the plane or in space. This vector shows both the direction and the distance of the overall change in position.

Vector Functions in Motion and Displacement

vector-functions-in-motion-and-displacement

introduction to vector functions

Vector Functions for Motion in Space

Vector functions describe motion in space by assigning a vector to each input value. In three dimensions, a vector function is often written as r(t) = . The component functions f(t), g(t), and h(t) give the position along the x-, y-, and z-axes. The variable t usually stands for time. That makes r(t) a position vector for a moving object. As t changes, the position vector traces a space curve C, which is the path of a point in three-dimensional space. The terminal points P = (f(t), g(t), h(t))...

space curves

Helices and Motion in Space Curves

Space curves show the path of an object moving through three-dimensional space. Unlike plane curves, which stay in two coordinates, space curves use three coordinate functions. If t is a parameter, the position of the particle is written as a vector function r(t) = . Here, x(t), y(t), and z(t) are differentiable functions of t. As t changes over an interval, the ends of the position vectors trace the curve. One common example is the circular helix, given by r(t) = . At t = 0, the position...

real-world applications of space curves

Using Space Curves to Predict Collisions

Space curves help predict whether two moving objects will collide in three-dimensional space. In aerospace navigation, radar systems track interceptors and moving aerial targets at the same time. Their flight paths are modeled mathematically so analysts can test for a possible impact within a given time interval. Each object is described by a vector function. A vector function gives the object’s position at each instant of time. Its x-, y-, and z-coordinate components show how the object moves...

derivatives of vector functions

Vector Motion and Instantaneous Velocity

Vector-valued functions describe position as a function of time. In Cartesian coordinates, a moving car on a curved road can be written as \begin{equation*} \textbf{r}(t)=\langle x(t),y(t),z(t)\rangle \end{equation*}. Each component gives the car’s location along one axis. The secant vector shows the change in position over a time interval. At time \(t\), the car is at point \(P\) with position \(\textbf{r}(t)\). After a short time \(h\), it reaches point \(Q\) with position...

integrals of vector functions

Vector functions describe motion in space when both speed and direction matter. A drone is a clear example, because its velocity changes as it moves. A vector-valued function assigns a vector to each time value. Each component is a real-valued function that tracks motion along one axis in space. These functions can model motion in two dimensions or three dimensions. In two dimensions, the velocity of the drone has two component functions. One component shows horizontal motion, and the other...

arc length of space curves

Distance Traveled Along a Space Path

Arc length gives the total distance traveled along a curve in space. For a moving object such as a helicopter, the path can be written as a vector-valued position function r(t) = , where t is time. This distance is not the same as displacement, which measures only the straight-line change between two points. Arc length includes every change in direction along the route. To find this length, the motion interval is split into many small time steps of size Δt. Over each step, the curve is treated...

curvature and its interpretation

How Curves Change Direction

Curvature measures how fast a curve changes direction at a point. A small curvature means the curve bends gently. A large curvature means it turns more sharply. For a space curve, the position of a moving object can be written as a vector-valued function r(t), where t often stands for time. The direction of motion comes from the tangent vector. The unit tangent vector is the tangent vector scaled to length 1, and it shows the instantaneous direction of travel along the curve. Curvature tells...

the normal and binormal vectors

Frenet Frame on a Helical Track

A helical track shows how tangent, normal, and binormal vectors describe motion in space. As a roller coaster car moves along the spiral, these three unit vectors stay perpendicular to one another. Together, they form the Frenet-Serret frame, a moving coordinate system for a curve in three dimensions. The unit tangent vector shows the direction the car is moving at that exact point. It comes from the first derivative of the position vector with respect to arc length. The unit normal vector...

torsion in vector calculus

Twisting Space Curves with Torsion

Torsion describes how a space curve twists out of the plane of bending. It works alongside curvature, which measures how sharply the path bends. Together, these ideas help describe motion in three dimensions. The Frenet–Serret frame gives a local picture of a curve at each point. It uses three unit vectors. The tangent vector points in the direction of motion, the normal vector points toward the center of curvature, and the binormal vector comes from the cross product of the tangent and normal...

motion in space: velocity and acceleration

motion-in-space-velocity-and-acceleration

Position Vectors, Speed, and Motion Changes

Motion in space can be described with a position vector r(t). This vector gives the location of an object, such as a drone moving through the air, at any time t. To study the motion, we look at how the position vector changes over time. Average velocity is found by dividing the change in position by the length of the time interval. When the interval becomes very small, this average velocity approaches a limit. That limit is the derivative of the position vector with respect to time. The...

projectile motion

Parabolic Paths in Projectile Motion

Projectile motion describes the path of an object launched into the air. A soccer ball kicked during a penalty is one example. In this model, air resistance is ignored, so gravity is the only force acting on the object. That assumption lets students treat the motion as two separate parts at the same time. The horizontal motion does not speed up or slow down, while the vertical motion keeps changing because gravity pulls downward. The initial velocity is a vector, which means it has both size...

tangential and normal components of acceleration

tangential-and-normal-components-of-acceleration

Curved Motion and Acceleration Vectors

Curved motion changes acceleration in two parts: one along the path and one across it. In particle motion, acceleration is often split into tangential and normal components so the change in velocity is easier to understand. The tangential direction follows the path of motion. It shows how the particle’s speed changes over time. The normal direction points toward the center of curvature, or the center of the bend in the path. It shows how the direction of motion changes. A particle moving along...

vector functions and motion: problem solving

vector-functions-and-motion-problem-solving

Drone Motion from Acceleration to Position

Drone motion can be tracked by using acceleration, velocity, and position vectors. In this problem, a UAV starts at position (3, 0, 0) and has an initial velocity in the positive z-direction. The motion is controlled by a time-dependent acceleration function a(t), which guides the drone during a precision inspection. To find the drone’s velocity, the acceleration vector is integrated over time. The initial velocity, v0 = (0, 0, vz), is used to set the constant of integration. This gives v(t) =...

key terms and concepts

Learn how vector-valued functions model drone motion, how integrating each component gives position, and how definite integrals find displacement.

Vector Functions in Motion and Displacement | 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.

calculus

Vector Functions in Motion and Displacement

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