Newton’s second law indicates that time derivative of a mass’ momentum is the net force on that object. Taking mass to be constant, the product of mass and acceleration is obtained.
$$ F=\frac{dp}{dt}=m\frac{dv}{dt} $$
Given that power is the product of force and velocity, we substitute the above and integrate to obtain an expression for kinetic potential energy.
$$ \frac{d\boldsymbol{\large\varepsilon}_K}{dt}=Fv\Rarr\int d\boldsymbol{\large\varepsilon}_K=\int Fv~dt=\int mvdv\Rarr\boldsymbol{\large\varepsilon}_K=\frac 12mv^2 $$
This derivation draws our attention to the fact that an instantaneous jump in velocity is impossible because it would require the application of infinite power.
An ideal spring generates a force proportional to its deflection from equilibrium position. Thus, the rate of change of force is proportional to the rate of change of deflection.
$$ F_s=kx\Rarr\frac{dF_s}{dt}\frac 1k=v $$
Again we apply conservation of energy, then substitute the above expression for velocity, and integrate to obtain an expression for spring potential energy. Substituting our initial definition further renders the normal expression.
$$ \frac{d\boldsymbol{\large\varepsilon}_S}{dt}=F_sv\Rarr\int d\boldsymbol{\large\varepsilon}_S=\int F_sv~dt=\frac 1k\int F_s~dF_s\Rarr\boldsymbol{\large\varepsilon}_S=\frac 1{2k}F_k^2=\frac 12kx^2 $$
This derivation draws our attention to the fact that an instantaneous jump in spring force is impossible because it would require the application of infinite power and an infinite speed of deflection.
A damper provides a resistive force related to velocity, and in the ideal case, this relationship is linear.
$$ F_{NLd}=f(v)~~~~F_d=bv $$
While a point mass may store kinetic potential energy, and a spring may store spring potential energy, a damper only dissipates energy.
When a point mass experiences a spring force and damping force, a second order one degree-of-freedom system is constituted. This system is only fully described by a velocity and position. Notably, the number of state-variables is equivalent to the number of energy storing components.
$$ m\ddot x+b\dot x+kx=0 $$