CLS-09

Work–energy theorem

W = ΔK = ½ m (v² − v0²).

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EnergyWork–energy theorem

Governing equation

W=\Delta K=\tfrac12 m(v^2-v_0^2)

$v \in \left[0, 100\right]$

where

m: Mass (kg)
v_0: Initial speed (m/s)
v: Final speed (m/s)
W: Net work (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

W=\Delta K=\tfrac12 m(v^2-v_0^2)

$v \in \left[0, 100\right]$

$v = 0$ $W = -18$

Along the curve

②

Domain of the variable — blue

v_{\min} = 0,\quad v_{\max} = 100

$v \in \left[0, 100\right]$

$v = 0$ $v = 0$

Along the curve

③

Response curve — green

W=\Delta K=\tfrac12 m(v^2-v_0^2)

$v \in \left[0, 100\right]$

$v = 0$ $W = -18$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}W}{\mathrm{d}v}

$v \in \left[0, 100\right]$

$v = 0.629$ $W^{\prime} = 2.516$

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-09 — Work–energy theorem) is the form associated with Work–energy theorem. Working symbols: $m$ , $v_0$ , $v$ $\rightarrow$ $W$ . The line integral of F·dr along a particle path equals the change in kinetic energy, the work–energy theorem.

Purpose

Purpose: compute

W

from

m

v_0

v

in Classical mechanics via

W=\Delta K=\tfrac12 m(v^2-v_0^2)

W = ΔK = ½ m (v² − v0²). 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 = 4.000\,\mathrm{kg}

v_0 = 3.000\,\mathrm{m/s}

v = 8.000\,\mathrm{m/s}

, the governing relation

W=\Delta K=\tfrac12 m(v^2-v_0^2)

yields

W = 110.000\,\mathrm{J}

. Particle, net work of all forces. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass m4 kgInitial speed v_03 m/sFinal speed v8 m/s

Outputs

Net work W110.000 J

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Narration of this film

Particle, net work of all forces.

The line integral of F·dr along a particle path equals the change in kinetic energy, the work–energy theorem.

Reading speed

Watch on YouTube