Question

Niagara Falls is about 50m high. What is the temperature rise in kelvins of the water from just before to just after it hits the rocks at the bottom of the falls, assuming negligible air resistance during the fall and that the water doesn't rebound but just splats onto the rock? It is helpful (but not essential) to consider a 1g drop of water.

Short answer

The temperature rises at 0.12 K .

Step-by-step solution

Definition of Thermal Energy

Thermal energy is the energy that controls the temperature of a system. The passage of thermal energy is what is referred to as heat. Thermodynamics is a discipline of physics that studies how heat is exchanged across systems and how work is performed throughout the process.

The thermal energy is given by$Q=mc∆T$ where mis mass of substance, $∆T$is change in temperature and cis specific heat.

Finding Rise in temperature

In the given case, thermal energy is change in energy so, replace Qby$∆E$in the formula of thermal energy and solve for $∆T$.

$$\begin{gathered} ∆E=mc∆T \\ ∆T=\frac{∆E}{mc} \end{gathered}$$

Substitute$∆E=mgh$and simplify.

$$\begin{gathered} ∆T=\frac{mgh}{mc} \\ =\frac{gh}{c} \end{gathered}$$

Substitute $g=9.8m{s}^{-2}$,$h=50\mathrm{m}$and $C=4.2\times {10}^3\mathrm{J}.{\mathrm{kg}}^{-1}.{\mathrm{K}}^{-1}$into the obtained formula to obtain the rise in temperature.

$$\begin{gathered} ∆T\mathit{=}\frac{\left(9.8{\mathrm{ms}}^{-2}\right)\left(50\mathrm{m}\right)}{4.2\times {10}^3\mathrm{J}.{\mathrm{kg}}^{-1}.{\mathrm{K}}^{-1}} \\ =0.12\mathrm{K} \end{gathered}$$

Therefore, the temperature rises at 0.12 K .