$$v_1 - v_2 = \frac{ds}{dt}$$

$$m_1 \cdot \frac{dv_1}{dt} = -K \cdot s$$

$$m_2 \cdot \frac{dv_2}{dt} = K \cdot s$$

$$\frac{dv_1}{dt} - \frac{dv_2}{dt} = \frac{d^2s}{dt^2}$$

$$\frac{d^2s}{dt^2} = -K(\frac{1}{m_1} + \frac{1}{m_2}) s$$

if

$$y^{(2)} = \kappa \cdot y$$

then

$$y = Asin(\sqrt{-\kappa} \cdot x)$$

$$s = A sin(\sqrt{K(\frac{1}{m_1} + \frac{1}{m_2})} \cdot t)$$

where A is a constant, meaning the maximum s that can achieve.

$$v_1 = v_0 + \frac{A}{m_1} \sqrt{\frac{K}{\frac{1}{m_1} + \frac{1}{m_2}}} \cdot cos(\sqrt{K(\frac{1}{m_1} + \frac{1}{m_2})} \cdot t)$$