The major thing you forgot is Inertia, I guess in your game the mass moves.
Gravity (9.8 m/s2) and spring force (compression or depression, proportional to the change of length) should be fine. Project that on x and y axes.
Then apply sum(forces) = mass x acceleration and you will now what is the acceleration of your mass, on x and y axes.
Then integrate, delta speed = dela time x acceleraton. Use a small constant for delta time and let the computer work, compute several positions per second, display only a few of them.
Then integrate again, delta postion = delta time x speed.
Finally, to manage the mass on the springs, split your springs in multiple small springs, each one having a mass and a straight shape. You will probably need to add a force (a torque in fact) that tend to keep two consecutive sections aligned.