The equation of motion for a M-S-D system may be written as
where is a position vector, is frictional drag, is spring force, and is the external force (if any). may have 1 or 2 or 3 dimension (or more, I suppose). We assume that the spring pulls the object directly toward the origin, with a force proportional to the distance from the origin, and that frictional drag is the direction oppostite to the velocity, and proportional to the magnitude of velocity. These assumptions can be written as and
Therefore we get the equation
In the 2-D case, we have, in the x-direction,
which looks totally normal. It involves the x-components of the spring force and the drag force. But here's where it gets tricky and then simple again. What are the equations for and ? It turns out the equations are
So we end up with
You get a similar result for the y-component:
So we see that for this special case - a linear mass-spring-dashpot system - the x- and y-parts do not affect each other, i.e. the equations of motion are totally separable. These two equations are what you need to solve. Since they are not coupled, you can solve them separately, in either order.