CLS-26

Escape speed

From a spherical mass.

Reading speed

GoverningEscape speed

Governing equation

v=\sqrt{2GM/r}

$r \in \left[1000, 4\times 10^{5}\right]$

where

M: M (Mearth)
r: r (km)
v: Escape speed (km/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.

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①

Governing law — red

v=\sqrt{2GM/r}

$r \in \left[1000, 4\times 10^{5}\right]$

$r = 1000$ $v = 28.23$

Along the curve

②

Domain of the variable — blue

r_{\min} = 1000,\quad r_{\max} = 4\times 10^{5}

$r \in \left[1000, 4\times 10^{5}\right]$

$r = 1000$ $r = 1000$

Along the curve

③

Response curve — green

v=\sqrt{2GM/r}

$r \in \left[1000, 4\times 10^{5}\right]$

$r = 1000$ $v = 28.23$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}v}{\mathrm{d}r}

$r \in \left[1000, 4\times 10^{5}\right]$

$r = 3509.4$ $v^{\prime} = -0.003$

Along the curve

Lecture brief

Historical brief

Newtonian mechanics (1687) plus energy, angular momentum, Kepler and the ballistic parabola remain the first language of motion. Every sheet here is a closed-form orbit, throw, spin or oscillator. This sheet (CLS-26 — Escape speed) is the form associated with Escape speed. Working symbols: $M$ , $r$ $\rightarrow$ $v$ . From a spherical mass. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

v

from

M

r

in Classical mechanics via

v=\sqrt{2GM/r}

From a spherical mass. Use it when a real classical mechanics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

M = 1.000\,\mathrm{Mearth}

r = 6371.000\,\mathrm{km}

, the governing relation

v=\sqrt{2GM/r}

yields

v = 11.186\,\mathrm{km/s}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

M M1 Mearthr r6371 km

Outputs

Escape speed v11.186 km/s

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

From a spherical mass. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube