Producing a Model of the Melting of Ice

Producing a model of the melting of ice

For this exploration I chose to model the melting of ice tea. My motivation started in maths class. It took me longer to understand calculus than most of my classmates and after I grasped it it still seemed abstract. This is when I set myself to apply what I was learning in some real life context, something that would help me understand it better. In the end, I decided to investigate the melting of ice tea.

With all the work that we get in school, I hardly ever found myself having a break. This is when my mother and I decided that every weekend we go out at the local Hema and help ourselves to a meal. While complaining about the difficulties of school, I always found that by the time I got to drinking my ice tea, it would have long melted. This is when I started thinking of how my ice tea melted. I immediately got the connection of what I had learnt the year before in my physics classes, in thermodynamics and in this case specifically about the latent heat of fusion, Q = mL and also specific heat capacity, Q = cmΔT. From this, I knew that the rate at which my ice tea was melting was very much influenced by the mass of ice I had to start with but also the difference in temperatures of the ice tea and the environment around it. Finally, it made sense that the more ice melted, the faster the rate of melting would be I.e since there is less ice to melt and also smaller difference with surrounding temperature. I therefore concluded that this was going to be a decreasing graph. With this I hoped to produce equations of the melting of ice and to create a model of this, which would allow me to calculate how long I could complain before my ice tea became water.

However my main goal, was to apply the calculus I was learning in class which then still seemed abstract to me, into a personal situation that could help me understand it better.

Collecting Data

To be able to create the model I wanted to create I had to solve a series of equations after performing an experiment. Seeing as I was focusing on the melting of ice, I decided to measure the mass of a melting ice block. As this ice block was a spherical shape, the starting equations are based off of this. To be able to graph the melting of the ice, I measured the starting mass of ice while in a room at room temperature and continued to measure its mass every 5 minutes for 1.5 hours. While this experiment could have taken a longer period of time to have gained a more accurate model, having done this would have not been practical. I also believe that the current data points are plenty enough to produce a reasonably accurate model of the melting of ice. From this model and the equation made thereof, the mass of the ice can be calculated at any given time. The collected data is on the following page.

Graphing

Using the collected data, I produced a scatter graph on Microsoft excel with time plotted on the X axis and mass plotted on the Y axis.

From this graph the following assumptions were made:

The constant is negative: the graph has a negative correlation, reflecting on that the rate of change of the radius is a negative, -k

When time (t) = 0, mass (M) = 790: the initial mass of the ice, the y-intercept is 790