CAL-11

Parametric speed

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

Reading speed

ParametricParametric speed

Governing equation

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

$\dot x \in \left[-20, 20\right]$

where

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

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

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

$\dot x \in \left[-20, 20\right]$

$\dot x = -20$ $v = 20.4$

Along the curve

②

Domain of the variable — blue

\dot x_{\min} = -20,\quad \dot x_{\max} = 20

$\dot x \in \left[-20, 20\right]$

$\dot x = -20$ $\dot x = -20$

Along the curve

③

Response curve — green

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

$\dot x \in \left[-20, 20\right]$

$\dot x = -20$ $v = 20.4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}v}{\mathrm{d}\dot x}

$\dot x \in \left[-20, 20\right]$

$\dot x = -19.75$ $v^{\prime} = -0.98$

Along the curve

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

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

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