gravity_actions_solved.txt

gravity_actions_solved.txt

Previously derived approximation:

let r(t) be the distance from the GravityAction at time t.
let f(t) be the force of the GravityAction at time t.

r'(t) = -f(t)/r(t)

With help from HyperRogue Discord's off-topic channel:

dr/dt = -f(t)/r(t)

r(t) dr = -f(t) dt

∫ r(t) dr = ∫ -f(t) dt

r²(t)/2 = -F(t) + C (where F(t) is an antiderivative of f(t))

r(t) = ±√(-2F(t) + C)

For convenience I'll ignore negative r(t)

r(t) = √(-2F(t) + C)

Let r₀ = r(0)

r(0) = √(-2F(0) + C) = r₀;
-2F(0) + C = r₀²
C = r₀² + 2F(0)

r(t) = √(-2F(t) + r₀² + 2F(0))