Physical interpretation of initial conditions for damped mass-spring system I have background in pure mathematics so my question is about physical meaning.
If we consider equation for damped mass-spring system, it is linear ordinary second order differential equation. So to get unique solution there must be se up two initial conditions. One from initial displacement of mass point and second for initial velocity of the mass on the spring. Now I need to connect it with physical world.
Suppose i have some real problem. I know that initial displacement is some exact value, I can measure it, but how can I get initial velocity?
 A: If you know the position at $t=\Delta t$ as well, you calculate the velocity by 
$$v=\frac{\Delta x}{\Delta t},$$
where $\Delta x$ is the displacement, you also mentioned.
Mathematically speaking we would $\lim_{\Delta t \to 0}$, but this is a good approximation if you do not take $\Delta t$ too large.
A: To get the initial conditions you guess and look at whether the guesses fit the physical situation. For example, suppose that you have a mass on a spring and you are holding mass so that the spring is slightly stretched and then release it. The mass is not moving at t=0 so the initial condition is that the velocity is zero at t=0. But you could imagine other situations. Suppose, for example, that the mass is resting on a piston and the piston pushes the mass up to some height and then drops away from the mass leaving it to move under the influence of the spring. Then the velocity of the mass at the first position at which the piston no longer has contact with the mass is the mass's initial velocity.
