CLS-32

Projectile range

Flat-earth vacuum range.

Reading speed

GoverningProjectile range

Governing equation

R=v^2\sin 2\theta/g

$th \in \left[5, 85\right]$

where

v: v (m/s)
th: th (deg)
R: Projectile range (m)

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

R=v^2\sin 2\theta/g

$th \in \left[5, 85\right]$

$th = 5$ $R = 28.32$

Along the curve

②

Domain of the variable — blue

th_{\min} = 5,\quad th_{\max} = 85

$th \in \left[5, 85\right]$

$th = 5$ $th = 5$

Along the curve

③

Response curve — green

R=v^2\sin 2\theta/g

$th \in \left[5, 85\right]$

$th = 5$ $R = 28.32$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}R}{\mathrm{d}th}

$th \in \left[5, 85\right]$

$th = 5.503$ $R^{\prime} = 5.588$

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-32 — Projectile range) is the form associated with Projectile range. Working symbols: $v$ , $th$ $\rightarrow$ $R$ . Flat-earth vacuum range. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

R

from

v

th

in Classical mechanics via

R=v^2\sin 2\theta/g

Flat-earth vacuum range. 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

v = 40.000\,\mathrm{m/s}

th = 35.000\,\mathrm{deg}

, the governing relation

R=v^2\sin 2\theta/g

yields

R = 153.263\,\mathrm{m}

. 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

v v40 m/sth th35 deg

Outputs

Projectile range R153.263 m

Reading speed

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Narration of this film

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

Flat-earth vacuum range. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube