B1  ^1.00 Determine the initial temperature $t_1$ of the ball if the icicle stopped descending when the ball had melted a hole of depth $H=10~\text{cm}$.

Let us consider that at some moment the temperature of the ball is $t$ and the depth of the chanell is $h$. Thermal balance:
$$
C(t_{1}-t)=\lambda m=\lambda \rho Sh, \quad (1)
$$where $m$ is the mass of fused ice.

B2  ^2.00 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.

The heat transfer power is proportional to the temperatures difference between the ball and the ice. The icicle's descent velocity is also proportional to the temperature difference:
$$
\lambda \rho Sv=\alpha(t-t_{0}), \quad (2)
$$where $\alpha$ is a heat transfer coefficient. The temperature $t$ can be expressed from equation (1) and substituted intto equation (2):
$$
\lambda \rho Sv=\alpha \left(t_{1}-\frac{\lambda \rho Sh}{C}-t_{0}\right), \quad \text{ откуда } \quad v=\alpha \left(\frac{t_{1}-t_{0}}{\lambda \rho S}+\frac{h}{C}\right).
$$Thus, the icicle's descent velocity depends linearly on the depth of the hole.

From the graph it's clear that $v_{0}=3v_{2}=0.3~\text{mm}/\text{s}$.