CAL-11

Parametric speed

v = √(ẋ² + ẏ²). Speed along a plane curve.

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ParametricParametric speed

Governing equation

v=\sqrt{\dot x^2+\dot y^2}

where

\dot x: x-velocity (m/s)
\dot y: y-velocity (m/s)
v: Speed (m/s)

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-11 — Parametric speed) is the form associated with Parametric speed. Working symbols: $\dot x$ , $\dot y$ $\rightarrow$ $v$ . The arc-length element is ds = v dt. If the parameter is time, v is the kinematic speed.

Purpose

Purpose: compute

v

from

\dot x

\dot y

in Calculus via

v=\sqrt{\dot x^2+\dot y^2}

v = √(ẋ² + ẏ²). Speed along a plane curve. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\dot x = 3.000\,\mathrm{m/s}

\dot y = 4.000\,\mathrm{m/s}

, the governing relation

v=\sqrt{\dot x^2+\dot y^2}

yields

v = 5.0000\,\mathrm{m/s}

. Plane parametric curve, enter ẋ and ẏ. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

x-velocity \dot x3 m/sy-velocity \dot y4 m/s

Outputs

Speed v5.0000 m/s

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Narration of this film

Plane parametric curve, enter ẋ and ẏ.

The arc-length element is ds = v dt. If the parameter is time, v is the kinematic speed.

Reading speed

Watch on YouTube