10925da7-1f46-4308-8853-c87aa7e7ca12 23 Валерий Очков Alvaro Diaz Falconi 4 5 false false 0 0 decimal

Гравитационный поезд из Москвы в Питер (Gravity Train)

Cauchy problem

eq m t x'' * m g * t x R / * - ≡ 0 x L 2 / - ≡ D\0028\x\0029\\0028\0\0029\ 0 ≡ 31 sys :

t x' t x t diff :

t x'' t x t 2 diff :

t x eq 1 el eq 2 el eq 3 el 31 sys dsolve rhs maple : L g R * sqrt t * R / cos * 2 / -

Check

eq 1 el m L * g * g R * sqrt t * R / cos * 2 R * / m g * L * g R * sqrt t * R / cos * 2 R * / ≡

0 x L 2 / -

0 x' 0

Travel time

τ t x 0 x - ≡ t solve maple : π R * g R * sqrt /

τ x L 2 /

τ x' 0

Values

R 6371 km * :

g g.e :

τ min 42.2029

θ φ Earth R θ sin * φ cos * R θ sin * φ sin * R θ cos * 31 mat :

Earth

Cities

Moscov

λ.M 55.76 ° * :

ϕ.M 37.62 ° * :

M 90 ° * λ.M - ϕ.M Earth :

Leningrad

λ.P 59.94 ° * :

ϕ.P 30.33 ° * :

P 90 ° * λ.P - ϕ.P Earth :

Distances

L M P - norme : km 633.1535

Rect line, gravity train lenght

Λ 2 R * L 2 R * / asin * : km 633.4143

Geodesic distance

Max Speed and acc

0.5 τ * x' km hr / 1413.9629

T τ 0 0.5 1 stack * :

k 13 range :

0 x'' g.e 0.0497

Position

1 km / x min * x * x min * 0 τ lele * T min / A k el T k el x 1 km / * : o augment 2 1 sys

Speed

hr km / x min * x' * x min * 0 τ lele * T min / A k el T k el x' hr km / * : o augment 2 1 sys

Acceleration

1 g.e / x min * x'' * x min * 0 τ lele * T min / A k el T k el x'' 1 g.e / * : o augment 2 1 sys

Cities

E Earth 0 0.5 π * 0 0.5 π * 3030 CreateMesh :

γ2 0.866 0.433 - 0.5 0.75 0 0.5 32 mat :

E γ2 * M transpose γ2 * o augment P transpose γ2 * o augment 3 1 sys

Alvaro

4 appVersion 1.0.8348.30405