Name: estimating distance from changing velocity
Uploaded: 2026-01-12T12:24:58.000Z
Duration: 1 min 29 s
Channel: JoVE

距离问题是利用不同时间点测量的速度来计算物体已经移动了多远。当速度变化时，可以通过添加小的位移间隔来近似总距离，每个间隔显示在短时间步长内的运动。例如，在比赛中，跑者在前三秒内持续加速。每半秒进行一次速度测量显示速度从0米每秒增加到6.2米每秒。这些测量数据通过将速度-时间图划分为半秒矩形，分别表示下和和，从而估算总距离。较低估计使用每个时间区间左端点的速度。将这些速度乘以半秒时间步长并将各结果相加，得到10.55米。另一方面，上估计使用右端点速度。将这些元素乘以时间步长并相加，得到13.65米。实际差距介于这两个估计之间。增加测量次数可以获得更准确的结果。对于无限次测量，距离等于曲线下的面积，由速度随时间的积分表示。

路程问题

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Integrals

积分

KEY TERMS AND CONCEPTS

关键术语与概念

Distance Problem

estimating distance from changing velocity

Estimating distance from changing velocity uses small time intervals and velocity values. When an object speeds up or slows down, its total displacement can be found by adding many short pieces of motion. This method gives an approximation first, and integral calculus gives the exact result later.

A runner who accelerates during the first three seconds of a race shows how the idea works. Velocity can be measured every half-second and multiplied by the 0.5-second time step. Using left-endpoint velocities gives an estimated distance of 10.55 meters. Using right-endpoint velocities gives a higher estimate of 13.65 meters.

These two estimates show that the true distance traveled should fall between them. More frequent velocity measurements make the estimate more accurate. To find the exact displacement, the sum of velocity times small time intervals is taken in the limit as the intervals get smaller and smaller.

This limit is the Riemann sum definition of an integral. It shows that total distance traveled is the integral of velocity over time. As the time steps become infinitesimally small, the summation matches the integral and removes the error from numerical approximation.

Estimating Distance from Changing Velocity

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key terms and concepts

Learn how changing velocity can be used to estimate distance with left- and right-endpoint sums, then find the exact result using limits and integrals.

Estimating Distance from Changing Velocity | 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

Estimating Distance from Changing Velocity

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