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I am trying to determine the position of a point mass $P$ moving in 2D space after time $t$. The system is defined as follows at time $t_0$:

  • $P$ has an initial position $\mathbf p_0 =(x_0,y_0)$
  • $P$ has an initial velocity $\mathbf v_0 =(v_{0_x},v_{0_y})$
  • $P$ has a fixed mass $M$
  • $P$ is attached to multiple ideal springs $S_1, S_2, ... S_n$
    • Each spring $S_i$ is attached at the other end to a fixed point $\mathbf q_i = (q_{i_x}, q_{i_y})$
    • Each spring $S_i$ has a different stiffness $k_i$
    • Each spring $S_i$ has a natural length $l_i$
    • Each spring $S_i$ may initially by stretched or compressed at $t_0$ by an arbitrary amount, so their initial lengths may not be equal to $l_i$

Here's an image to illustrate.

$P$ and springs

This is tricky to compute in one go since as $P$ moves, its velocity and acceleration change due to the springs changing in length. However, for any possible position for $P$, the acceleration vector is known and fixed, since the $\mathbf q_i$ and spring parameters are fixed.

I understand that at $t_0$ I can compute the 2D acceleration vector of $P$ by just summing the force vectors produced by each spring $S_i$ towards or away from each $\mathbf q_i$. My intuition tells me that to compute the final position of $P$ at time $t$, I need to integrate this 2D acceleration over time twice. once to get the velocity delta/acceleration, which I can add to $\mathbf v_0$, and once again to get the displacement, which I can add to the initial position.

Assuming this is a viable approach, the problem I'm running into is that the $x$ and $y$ components for the spring lengths (and hence forces) are related. Is this a multi-variable calculus/vector field problem? How can I write an equation for the position of $P$ at time $t$?

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closed as off-topic by sammy gerbil, ZeroTheHero, Yashas, Kyle Kanos, Jon Custer Apr 10 '17 at 13:34

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – sammy gerbil, ZeroTheHero, Yashas, Kyle Kanos, Jon Custer
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Does your question mean you are to simulate this in a computer, or are you finding the exact analytical equation to solve the position at time t? $\endgroup$ – philip_0008 Apr 9 '17 at 20:15
  • $\begingroup$ I would like to obtain the exact position for a non-zero, arbitrary time step. In a computer simulation, this could be approximated by assuming the acceleration is constant for a given position for $P$ and computing several small time steps, modifying $\mathbf v$ by the acceleration times time elapsed each time. $\endgroup$ – user1832602 Apr 10 '17 at 0:44
  • $\begingroup$ How familiar are you with Lagrangian formulation of mechanics? $\endgroup$ – Kyle Kanos Apr 10 '17 at 10:03
  • $\begingroup$ Not at all, but perhaps that's a good place to start. To be clear, would this be the only path forward for solving this problem exactly? Also, I see that the question has been marked as off-topic. Fair enough, but I really would like to solve this problem. Did it fail the lack of effort check or is it not specific enough? The problem with the latter, of course, being that I don't know which specific physics concept needs to be used to solve this! $\endgroup$ – user1832602 Apr 11 '17 at 2:44
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    $\begingroup$ Even with only two springs the problem has no analytical solution except for small displacements from equilibrium. Numerical solution is the only way you will do this. $\endgroup$ – sammy gerbil Jun 10 '18 at 21:11