CLS-07

Level projectile range

R = v² sin(2θ) / g on level ground.

Reading speed

ProjectilesGalileo / Tartaglia

Governing equation

R=\dfrac{v^2\sin 2\theta}{g}

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

where

v: Muzzle speed (m/s)
\theta: Launch angle (°)
R: Range (m)
H: Max height (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=\dfrac{v^2\sin 2\theta}{g}

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

$\theta = 5$ $R = 28.32$

Along the curve

②

Domain of the variable — blue

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

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

$\theta = 5$ $\theta = 5$

Along the curve

③

Response curve — green

R=\dfrac{v^2\sin 2\theta}{g}

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

$\theta = 5$ $R = 28.32$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}R}{\mathrm{d}\theta}

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

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

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-07 — Level projectile range) is the form associated with Galileo / Tartaglia. Working symbols: $v$ , $\theta$ $\rightarrow$ $R$ , $H$ . Galileo's superposition of uniform horizontal motion and free fall integrates to a parabola whose range peaks at 45°.

Purpose

Purpose: compute

R

H

from

v

\theta

in Classical mechanics via

R=\dfrac{v^2\sin 2\theta}{g}

R = v² sin(2θ) / g on level ground. 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}

\theta = 35.000\,\mathrm{^{\circ}}

, the governing relation

R=\dfrac{v^2\sin 2\theta}{g}

yields

R = 153.263\,\mathrm{m}

H = 26.829\,\mathrm{m}

. No drag, flat Earth, launch and land at the same height. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Muzzle speed v40 m/sLaunch angle \theta35 °

Outputs

Range R153.263 m
Max height H26.829 m

Reading speed

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Narration of this film

No drag, flat Earth, launch and land at the same height.

Galileo's superposition of uniform horizontal motion and free fall integrates to a parabola whose range peaks at 45°.

Reading speed

Watch on YouTube