CLS-14

Mechanical energy

E = ½ m v² + m g h. Conserved when only gravity does work.

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NewtonianWork–energy

Governing equation

E=\tfrac12 mv^2+mgh

$v \in \left[0, 40\right]$

where

m: Mass (kg)
v: Speed (m/s)
h: Height (m)
E: Energy (J)

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

E=\tfrac12 mv^2+mgh

$v \in \left[0, 40\right]$

$v = 0$ $E = 58.86$

Along the curve

②

Domain of the variable — blue

v_{\min} = 0,\quad v_{\max} = 40

$v \in \left[0, 40\right]$

$v = 0$ $v = 0$

Along the curve

③

Response curve — green

E=\tfrac12 mv^2+mgh

$v \in \left[0, 40\right]$

$v = 0$ $E = 58.86$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E}{\mathrm{d}v}

$v \in \left[0, 40\right]$

$v = 0.252$ $E^{\prime} = 0.503$

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-14 — Mechanical energy) is the form associated with Work–energy. Working symbols: $m$ , $v$ , $h$ $\rightarrow$ $E$ . Kinetic plus gravitational potential. Other potentials add their own terms.

Purpose

Purpose: compute

E

from

m

v

h

in Classical mechanics via

E=\tfrac12 mv^2+mgh

E = ½ m v² + m g h. Conserved when only gravity does work. 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 = 2.000\,\mathrm{kg}

v = 4.000\,\mathrm{m/s}

h = 3.000\,\mathrm{m}

, the governing relation

E=\tfrac12 mv^2+mgh

yields

E = 74.9\,\mathrm{J}

. A mass sliding a ramp, two energy bars. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass m2 kgSpeed v4 m/sHeight h3 m

Outputs

Energy E74.9 J

Reading speed

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Narration of this film

A mass sliding a ramp, two energy bars.

Kinetic plus gravitational potential. Other potentials add their own terms.

Reading speed

Watch on YouTube