Hello everyone! Since this is my first post as one of the virtuosi, I should probably introduce myself a little. I'm a first year graduate student in physics at Cornell university. I did my undergraduate work at Oberlin college (I know, you've never heard of it), and I'm currently just trying to keep my head above water and take in as much physics as I can. Additionally, I'm trying to find work for the summer, I might post more on that later.

Today, the question that is on my mind is: How much does a water droplet heat up when it goes over niagara falls?

(image from http://grandcanyon.free.fr/)

Let's begin with a little motivation. Why would a water droplet heat up when it falls? Well, the physical mechanism is that as it falls through air, air resistance dissipates energy. This energy is dissipated mostly as heat, so we expect to put some additional heat into our water droplet as it falls.

More quantitatively, assume the water is going to fall from some height h to the ground. The gravitational potential energy of our water droplet is given by

\[ PE=mgh \]

where m is the mass of the droplet and g is acceleration due to gravity. We assume that the particle starts with no initial velocity. It is very easy to place an upper bound on how much the droplet will warm. The maximum heating will happen if all of the potential energy were converted to thermal energy. The temperature change wound be

\[ mc\Delta T = mgh \]

so

\[ \Delta T = \frac{gh}{c} \]

Where c is the specific heat of water.

Niagara falls is 51m tall, g is 9.8m/s*s and c is 4.1kJ/kg*K so this gives a maximum temperature change of .12C=.22F, fractions of a degree.

However, we can do better than this.