# Assume put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit...

## Question:

Assume put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on the table. Describe how the temperature of the water changes as time passes.

## Newton's Law of Cooling

One important result that Isaac Newton defined involved the temperature of objects and their surroundings. Specifically, he theorized that an object at a different temperature will asymptotically approach the temperature of its surroundings as time passes. This temperature change is exponential, and the temperature will change more rapidly at first before approaching the horizontal asymptote of the surrounding temperature.

This temperature change is defined using a differential equation, which is defined as Newton's Law of Cooling. This is because the rate of change of the temperature depends on two factors: how long the glass of water has been sitting out and the original temperature difference between the ice and the water.

At first, the temperature will change quickly, as the ice and water wish to be at the same temperature. However, as the ice warms up to to the temperature of the water, there's less and less of a difference between the temperature of the ice and the temperature of the water. Thus, this rate of change of the temperature will slow down. The overall temperature of the ice and water will approach an equilibrium value asymptotically, as this type of change represents an exponential function.

{eq}\frac{dT}{dt} = k(T - T_w) {/eq}

Note that this mathematical model assumes that the amount of water is large enough in comparison to the amount of ice that we are "warming up" the ice, not "cooling down" the water. We assume that the temperature of the water itself changes only a negligible amount. As this is usually not the case, this model would need to be modified, which would make it greatly more complex.