Observations

We derived that the velocity of the mass-string contact point is

$$v(t) = v_0\, e^{-{\frac{2R}{m}t}}, \quad t\ge 0$$

• At time zero the mass velocity is $v_0$, as it must be
• Velocity decays exponentially to zero with time-constant $m/2R$
• Decay rate is proportional to $R/m$
• Since $R=\sqrt{K\epsilon}$, decay is faster if either string tension $K$ or mass-density $\epsilon$ is increased
• String displacement at $x=0$ is given by the integral of transverse velocity:
  $$y(t,0) = \int_0^t v(\tau)\,d\tau = v_0\,\frac{m}{2R}\,\left[1-e^{-{\frac{2R}{m}t}}\right]$$
  where we define initial displacement to be $y(0,0)=0$