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}

where

v: Muzzle speed (m/s)
\theta: Launch angle (°)
R: Range (m)
H: Max height (m)

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

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

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