CLS-12

Escape velocity

v_esc = √(2 G M / R) from a spherical mass.

Reading speed

GravityNewton

Governing equation

v_{\mathrm{esc}}=\sqrt{\dfrac{2GM}{R}}

$R \in \left[10, 1\times 10^{6}\right]$

where

M: Mass (kg)
R: Radius (km)
v_{\mathrm{esc}}: 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_{\mathrm{esc}}=\sqrt{\dfrac{2GM}{R}}

$R \in \left[10, 1\times 10^{6}\right]$

$R = 10$ $v_{\mathrm{esc}} = 282.3$

Along the curve

②

Domain of the variable — blue

R_{\min} = 10,\quad R_{\max} = 1\times 10^{6}

$R \in \left[10, 1\times 10^{6}\right]$

$R = 10$ $R = 10$

Along the curve

③

Response curve — green

v_{\mathrm{esc}}=\sqrt{\dfrac{2GM}{R}}

$R \in \left[10, 1\times 10^{6}\right]$

$R = 10$ $v_{\mathrm{esc}} = 282.3$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}v_{\mathrm{esc}}}{\mathrm{d}R}

$R \in \left[10, 1\times 10^{6}\right]$

$R = 6299.2$ $v_{\mathrm{esc}}^{\prime} = -0.022$

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-12 — Escape velocity) is the form associated with Newton. Working symbols: $M$ , $R$ $\rightarrow$ $v_{\mathrm{esc}}$ . Setting total mechanical energy to zero on a 1/r potential gives the speed that just reaches infinity, √(2GM/R).

Purpose

Purpose: compute

v_{\mathrm{esc}}

from

M

R

in Classical mechanics via

v_{\mathrm{esc}}=\sqrt{\dfrac{2GM}{R}}

v_esc = √(2 G M / 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 = 5.972e+24\,\mathrm{kg}

R = 6371.000\,\mathrm{km}

, the governing relation

v_{\mathrm{esc}}=\sqrt{\dfrac{2GM}{R}}

yields

v_{\mathrm{esc}} = 11.186\,\mathrm{km/s}

. Non-rotating sphere, no atmosphere, Newtonian. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass M5.972e+24 kgRadius R6371 km

Outputs

Escape speed v_{\mathrm{esc}}11.186 km/s

Reading speed

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Narration of this film

Non-rotating sphere, no atmosphere, Newtonian.

Setting total mechanical energy to zero on a 1/r potential gives the speed that just reaches infinity, √(2GM/R).

Reading speed

Watch on YouTube