W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity

Eliminating Transcendental Angles from Ballistics, Wave Refraction, and Lorentz Boosts

A Galilean projectile launched from O with initial speed-squared v^2 and elevation spread s traces a parabola whose apex height h = v^2*s/(2*g), horizontal range quadrance Q_range = 16*(v^4/(4*g^2))*s*(1 - s) = (v^4/g^2)*S_2(s), and focus-directrix geometry depend rationally on s; Snell’s refraction law becomes n_1^2 * s_1 = n_2^2 * s_2 in spreads.

Interactive GEOMETOR construction for W.26 (Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point V_apex : Point R_range : Point ===== traj = projectile_parabola(O, V_apex, R_range) F_traj = focus(traj) l_dir = directrix(traj) l_iface = line(I_inc, P_ref) l_norm = altitude(l_iface, P_ref)

Mathematical Synthesis

Physics formulas are filled with \(\sin^2\theta\), \(\cos^2\theta\), and \(\sin(2\theta)\) that become purely algebraic when expressed in quadrance and spread (Wildberger2005). For a projectile launched from O landing at Rrange via apex Vapex along trajectory traj (focus Ftraj, directrix ldir), the horizontal range quadrance is governed directly by the logistic spread polynomial \(S_2(s) = 4 s (1 - s)\), immediately proving maximum range at \(s = 1/2\) and equal ranges for complementary spreads \(s\) and \(1 - s\)!

Across an optical interface liface with normal lnorm at Pref (incident from Iinc, refracted toward Ttrans), Willebrord Snellius’ law \(n_1 \sin\theta_1 = n_2 \sin\theta_2\) squares cleanly to the linear spread proportion \(n_1^2 s_1 = n_2^2 s_2\), with total internal reflection occurring precisely when \((n_1^2 / n_2^2) s_1 > 1\)! And in special relativity, Einstein’s velocity addition and spacetime intervals are native instances of the Red Minkowski metric (W.23).

Key Algebraic Identities

\[y = x \sqrt{\frac{s}{1-s}} - \frac{g x^2}{2 v^2 (1 - s)}, \qquad Q_{\text{range}} = \frac{v^4}{g^2} S_2(s) = \frac{4 v^4}{g^2} s(1 - s)\]

\[\text{Snell's Spread Law:}\quad n_1^2\, s_1 = n_2^2\, s_2, \qquad s_{\text{crit}} = \frac{n_2^2}{n_1^2}\; (n_1 > n_2)\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Projectile range law Q_range(s) ~ S_2(s)	S_2(1/2) = 1, S_2(s) == S_2(1 - s)	Complementary spreads land identically
Trajectory focus & directrix	F_traj = [4, 0], l_dir: y = 4	Exact focus-directrix parabola
Snell's Spread Law n_1^2 * s_1 == n_2^2 * s_2	9 * (16/25) == 16 * (9/25) == 144/25	Integer ray vectors [-4,3] -> [3,-4]

Cross-Repository Canon

Executable Demonstration	W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity — Code (Standalone model.py & w-26.svg)
Thematic Part	Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves
Physical Canon	Galileo Galilei (1638) & Willebrord Snellius (1621)
GEOMETOR Fundamentals	F.1 Albert’s Parabola, F.22 Functions
Chromogeometry Bridge	Relativistic spacetime intervals instantiate the Red metric Q_R (W.23)
Optics Bridge	Complements parabolic mirror reflection in W.21

Constructive Dependency Graph

Depends on: W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads, W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas, W.21 Taylor Conics, Algebraic Tangents & Optical Reflection Laws, W.23 Chromogeometry & Universal Symmetric Bilinear Forms