<p> Golden ratio </p>

5b746eaf-d1a8-4b1f-813b-072b62ef0522 12 15 12 decimal

&[DATE] &[TIME] - &[FILENAME]

m 189 :

V 17 :

p 1.42 :

I.xG 6.73 :

g 9.8 :

h.G 0.53 :

I.zG 36.4 :

ω 0.215 :

b.G 0.703 :

t x 501 ω t * π 2 / + sin - (* : t y 100 ω t * sin * : 21 sys

curvatura traiettoria

t c t x t diff t y t 2 diff * (t y t diff t x t 2 diff * (- (abs t x t diff (2^t y t diff (2^+ (32 /^/ :

x 0 > ( x 9 < ( & x c not defined if

raggio di curvatura

t R.c 1 t c / :

x 0 > ( x 9 < ( & x R.c not defined if

t α b.G t R.c / atan :

t Δ p t R.c / atan :

t F.sf b.G p t Δ cos * / m * V 2^t R.c / * :

t ψ'' m V 2^* b.G * t R.c / t α cos * t F.sf p * t Δ cos * - I.zG m b.G 2^* + / eval :

t ψ'' i.e: the RHS is the 2nd derivative ≡

If you want the 1rst derivative: simply integrate

t 012 0.05 range : data t t ψ'' vectorize augment : 21 sys

x λ 10 x ≤ (x 12 ≤ (& cases :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

X data Σ 1 col : Y data Σ 2 col : 12 mat x q X Y x linterp : 21 sys

u 7.5 :

u q 0.006837279604925 -

x ψ'' x λ * data Σ u u q + 20 black 1 5 mat 3 1 sys

t Clear 1

6 min @ acc 50 x f t ψ'' t 0 x int :

x f x λ * <p> Golden ratio </p>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

x f x ψ'' :

minX f 78 y1 y2 ≤ 1012 -^Extrema 1 el : 7.30602927377092

minX f 0.004675467442971 -

∫ x 012 0.25 range : i 1 x rows range cumul i el t ψ'' t x 1 el x i el int : for x cumul augment 31 line :

∫

t0 1 time :

x Y D x ψ'' :

init 0 :

end 12 :

x λ 10 x ≤ (x end ≤ (& cases :

sol init 0 end 240 x Y D dn_GearsBDF :

x ψ'' x λ * sol 2 1 sys

1 time t0 - 5 s *

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 Y i 1 + el Y i el - X i 1 + el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X 1 deriv * augment 101 line :

sanity the project

sol Deriv