Question

The temperature right above the lake was 5°C, Your Grandpa said the lake wont crack, it is 65 feet deep and the temperature at the bottom of the lake is 20°C and he wanted to walk on the lake and start ice fishing. You didnt know if you should listen to Grandpa so you needed to calculate the thickness of the ice to see if it can hold. Your mass is 80 kg and your grandpas mass is 54 kg. Was Grandpa right?

Answer 1

This is a very interesting question, to model ice thickness one of the most used equations is stefan's equation, we can start with the heat equation and some boundary conditions on the temperature at the top of the slab of sea ice and the bottom of the slab of ice. The detail here is that the ice thickness depends on the time and the number of days at a certain temperature

since time is not given let's assume a thickness per 1 time interval

ρi is ice density, which is about 0.9 grams/cm^3, L is the latent heat of ice (335 Joules/gram), the amount of energy it takes to heat ice one degree, κi is the heat conductivity of ice, and q depends on the solar radiation penetrating into the ice. A good approximation is

65 ft=1981.2 cm

To estimate the ice load we can use  the Gold’s Formula provides an estimate of the load limit for a particular thickness of ice, below which the failure of an ice sheet is unlikely. It is not an infallible measure of the bearing capacity of an ice sheet and must be combined with field observations of other factors affecting ice strength when actual load limits are determined.

Where A is a constant that depends on the type of ice. A=4 is generally used for models, P is in kilograms, now the requiered thickness to support the persons is

You would need an ice thickness of 5.78 cm to support both persons, if the ice is formed just for 1 day that 0.041 is the ice growth per day, we would need 5.78/0.041=140.97 days. 141 days of ice growth at that temperatures to support the weights.

This is a simple model of a situation that is much more complex because it depends on time, amount of solar light on the lake, geographic location, the type of ice, the A factor of gold formula. I hope this approximation results to be helpful