A ball of very high thermal conductivity is attached to a thread. The thread goes through a thin vertical hole in a cylindrical icicle, which coincides with its axis. At the begining, the ball is heated up to the temperature $t_1$, and the temperature of icicle is equal the ambient temperature $t_0 = 0^\circ\text{C}$. Due to the melting of ice, the icicle moves downwards, and the meltwater flows out as drops at temperature $t_0$.

A cylindrical hole of cross-sectional area $S=2~\text{cm}^2$ is formed behind the ball.

B1 Determine the initial temperature $t_1$ of the ball if the icicle stopped descending when the ball had melted a hole of depth $H=10~\text{cm}$.

A S

B2 At the moment when the icicle had descended by $2H/3$ its velocity was $v_2 = 0.1~\text{mm}/\text{s}$. Determine the velocity $v_0$ of the icicle at the initial stage of the experiment.

A S

Assume that the heat transfer power is proportional to the temperature difference between the ice and the ball, and that all of it goes into melting of ice. The heat capacity of the ball is $C=59.6~\text{J} \cdot^\circ\text{C}^{-1}$. The specific heat of melting of ice is $\lambda=330~\text{kJ} \cdot \text{kg}^{-1} $. The density of ice is $\rho=900~\text{kg} \cdot \text{m}^{-3}$.