I’ve even got a starter question to get you guys into the scenario.
Once you’ve completed the starter question, under the solution comment attaches the main question, which is unsolved.
Not sure how much you’d care, but I came back to this and found (by hand) a function which closely approximates the solution - it’s not exact but it’s also not super far. Graph.
I think I could also solve the differential equation by hand at this point (getting the same solution as before) - I haven’t, but I’m pretty sure I could if I wanted to for whatever reason. I’m doubtful it’s possible to get an exact solution in terms of just x but if you ever manage I’d love to see how.
Kind-of solution
F = ma and m = 1, so a = F = 1/r^2. I used a = -1/r^2 so the force experienced by the particle at r = 5 would be negative. a = dv/dt.
dv/dt = -1/r^2
v dv/dt = -1/r^2 dr/dt → v = dr/dt. Multiply both sides by v, but write it as dr/dt on the right
v dv = -1/r^2 dr → Multiply both sides by dt
0.5v^2 = 1/r + C → Integrate
v^2 = 2/r + C → Multiply both sides by 2, find C = -2/5 by plugging in r = 5, v = 0
v^2 = 2/r - 2/5
5*arcsin(sqrt(r/5)) - sqrt(5r-r^2) = sqrt(2/5) t + C → Solve the differential equation y’^2 = 2/y - 2/5 with Wolfram Alpha, find C = 5π/2 by plugging in r = 5, t = 0
5*arcsin(sqrt(r/5)) - sqrt(5r-r^2) = sqrt(2/5) t + 5π/2 → Can find that the particle reaches the origin at t = 5sqrt(5)π / 2sqrt(2) ≈ 12.418 seconds
I made heavy use of internet resources on this one, not just for solving the differential equation but also for technique - I got a bunch of wrong answers before getting this one that I finally think is right, at least up until the moment of origin contact
Graph (the hidden ellipse is because I thought the path may be elliptical, but it doesn’t appear to be)
Solution (starter question):
spoiler
Please refer to the main post, if you don’t like looking at the image. https://gmtex.siri.sh/fs/1/School/Extra/Maths/Unsolved/1d-gravity.html
For the main question, you are encouraged to share your progress
spoiler
You might be able to solve this with differential equations, or by solving the iterative functions, I dont know