How far down can a nickel ball of radius r and temperature T_{n} melt into the ice block of temperature T_{i}? Watch the video and state your assumtpions...

The distance should depend on the thermal diffusivity of the ball (κ), the temperature difference between the ball and ice (##\Delta T = T_n - T_i##), the gravitational acceleration constant (g), the specific latent heat of melting for ice (L), the thermal conductivity of the ice (λ), and the radius of the ball (r). A combination of these variables that has the dimension of length and is physically plausible is

## d = \frac{\Delta T λ r^4 g}{κ^2 L} ##

The distance may also involve the dimensionless ratio of the density of the ball to that of the ice.

Assume a ball of 1 cm^{3}. The specific heat of nickle is 0.44 J/gK and its density is 8.9 g/mL, so it should release 3.9 J for each degree of Celcius cooled. The heat of fusion of ice is 335 J/g and its density is 0.934 g/mL, so melting 1 mL of ice requires 312 J of energy. So, a ball ideally should be able to melt its own volume of ice for every 80^{o}C above 0^{o}C. Based on blackbody radiation curves, red hot objects tend to be ~600^{o}C, which would correspond to melting ~ 7-8x the volume of the ball. In the video, the ball melts ~ 3-4x its size, indicating that a lot of the heat is lost (for example, you see a lot of steam produced, meaning that not all of the energy goes into melting the ice, and a good deal of thermal energy is probably lost when the water pours out of the side of the ice block).