Why is the damping force acting on an oscillating system opposite in direction to velocity and not acceleration? So far I know that the damping force is a frictional force that opposes motion and so it acts in the opposite direction to velocity . Bit why can't the same be said for acceleration doesn't the damping force reduce acceleration too? Thankyou for the help
 A: The equations of motion for the position determine the accelerations: they are second-order differential equations in time:
$$\vec F = m\vec a = m\ddot{\vec x }$$
So the acceleration, the second derivative of the location in time, has to be determined from the state of the physical system in some way. Typically, it is determined using the $F=ma$ formula above: the acceleration is the total force divided by the mass.
For this law to actually imply some trajectory, the force has to be calculable from the current properties of the particle. But the properties only include the location and its derivative, the velocity. They don't include the acceleration itself which is indeed what we want to calculate.
If the force were defined as a function of the accelreation, we would get a strange self-referring equation $a=f(a,\dots)$. If there were no other position- or velocity-dependent force, the solution of this equation would be a "universal" value of the acceleration, a very strange situation.
So the forces may only depend on locations and their first derivatives, the velocities.
In particular, friction forces are typically parallel to the velocities, going in the opposite diretion. $-\epsilon\cdot \vec v$ is the best direction in which the velocity should be changed in $dt$, i.e. the best direction of the force and the acceleration, that achieves the same reduction of the kinetic energy with a minimum value of $|\Delta \vec v|$.
Alternatively, one may say that $\vec v$ is the only direction that the force may have if the laws are translationally invariant. That condition implies the independence of the force on $\vec x$. The velocity is the only thing that the force may depend upon and the rotational symmetry implies the proportionality to the velocity. The coefficient is negative because by the second law, the kinetic energy has to dissipate, not get concentrated.
One may also derive the force microscopically. Imagine some gas with lots of particles that have random velocities. What is the total force these atoms exert on a solid? The forces from atoms nearly cancel but the atoms hitting from the front, in the direction of the velocity, are a bit more frequent and have a bit higher relative velocity thatn those in the opposite direction. So they slightly dominate, making $\vec a\sim -\vec v$.
