Kerepakupai Vena in Venezuela, more commonly known as Angel Falls, is the world\

Question

Kerepakupai Vena in Venezuela, more commonly known as Angel Falls, is the world's highest waterfall with a height of 979 m and 47 different plunges. The longest plunge (for the water) is 807 m. At the bottom of the fall, the kinetic energy of the water is converted into thermal energy. This results in a temperature increase of the water. If the water has a temperature of 17.1°C when it gets to the bottom of the plunge, what was the temperature at the top of the plunge? The specific heat of water is 1.00 kcal/(kg · °C).

Explanation / Answer

None! The gravity potential is initially converted to kinetic energy. Before the water reaches the bottom, its rapid motion turns it into a spray, and air resistance slows the fall converting much of the kinetic energy to heat. Evaporation cools the spray, perhaps removing more heat than what resulted from the loss of gravity potential. So the water at the bottom may even be colder than at the top.

For all of the gravity potential to be retained by the water as heat, you would have to run the water thru an adiabatic pipe with baffles to control the speed on the way down

A more realistic expectation is that PE --> ke + qe; where some of that potential energy PE = mgh converts to heat energy qe = m c dT , but most of it becomes kinetic energy ke = 1/2 mv^2. After all, it's the fall (velocity) of the water that makes it a waterfall. There has to be kinetic energy when the mass of the water has velocity just prior to impact 100 meters below its starting point.

Thus the way to find qe is to note that it is the difference between PE - ke = KE - ke = qe. KE = 1/2 mV^2 is the kinetic energy of the waterfall just prior to impact if all the PE were converted into kinetic energy. Thus we have...

qe = m c dT = 1/2 m(V^2 - v^2) and the change in temperature (K) dT = 1/2 (V^2 - v^2)/c where c is the water specific heat, V is the max possible velocity found from V^2 = 2gh where h = 100 m, m is the mass of the water, and v is the measured actual velocity at the bottom of the falls. So you need to know the actual waterfall velocity v to determine how much of the PE is converted into heat.

Note, if V = v, there are no losses in energy when all the PE is converted into KE. Thus dT = 0 because KE - KE = qe = 0. If all the PE were converted into heat, there would be no falls as there would be no kinetic energy.

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