CLS-03

Linear momentum

p = m v, and F = dp/dt.

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MomentumNewton 1687

Governing equation

p=mv,\quad F=\dfrac{\mathrm{d}p}{\mathrm{d}t}

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

where

m: Mass (kg)
v: Velocity (m/s)
p: Momentum (kg·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.

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①

Governing law — red

p=mv,\quad F=\dfrac{\mathrm{d}p}{\mathrm{d}t}

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

$v = -100$ $p = -300$

Along the curve

②

Domain of the variable — blue

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

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

$v = -100$ $v = -100$

Along the curve

③

Response curve — green

p=mv,\quad F=\dfrac{\mathrm{d}p}{\mathrm{d}t}

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

$v = -100$ $p = -300$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}p}{\mathrm{d}v}

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

$v = -98.74$ $p^{\prime} = 3$

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-03 — Linear momentum) is the form associated with Newton 1687. Working symbols: $m$ , $v$ $\rightarrow$ $p$ . Newton's 'quantity of motion' is mass times velocity. Its time derivative is the net force, so p is conserved when F = 0.

Purpose

Purpose: compute

p

from

m

v

in Classical mechanics via

p=mv,\quad F=\dfrac{\mathrm{d}p}{\mathrm{d}t}

p = m v, and F = dp/dt. 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 = 3.000\,\mathrm{kg}

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

, the governing relation

p=mv,\quad F=\dfrac{\mathrm{d}p}{\mathrm{d}t}

yields

p = 24.000\,\mathrm{kg·m/s}

. Particle of constant mass. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mass m3 kgVelocity v8 m/s

Outputs

Momentum p24.000 kg·m/s

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Narration of this film

Particle of constant mass.

Newton's 'quantity of motion' is mass times velocity. Its time derivative is the net force, so p is conserved when F = 0.

Reading speed

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