## The Growth Rate of Ice on a Pond

Ice will start to form on a pond when the temperature of the air falls below 0 ° C. Suppose the air temperature is θ° C and the water temperature is 0 ° C. Take the thickness of the ice to beand suppose that in a time dt the thickness increases byThe heat released as the ice freezes has to escape through the layer of ice that has already formed.

The heat released by the water as a massturns to ice iswhereis the specific latent heat of fusion of water. If the density of ice isthenwhere is the volume of ice formed andis the area of the ice, so the energy given up is

If this freezing happens in a timethe amount of heat that escapes through the ice isEquating the two expressions gives

Cancellingand rearranging gives

We can separate variables and integrate. Takewhen

Sincewhenwe havehence