Answer:
The water temperature at the bottom of the 200-meter high waterfall is calculated to be 20.47°C. This represents a rise of 0.47°C from the initial 20°C at the top. The increase results from the conversion of potential energy into kinetic energy during the fall and its subsequent conversion into thermal energy upon impact, illustrating the principles of energy conservation and transformation.
Explanation:
The water's potential energy at the top of the waterfall transforms into kinetic energy as it falls. This kinetic energy, at the point of impact, converts partly into thermal energy, increasing the water's temperature. The velocity of water at the bottom of the waterfall is crucial for calculating this temperature increase. We use the formula:
[tex]$\begin{align*}v &= \sqrt{2gh} \\[6pt]\end{align*}[/tex]
where:
Inserting these values, we find:
[tex]$\begin{align*}v &= \sqrt{2 \times 9.81 \times 200} \\[6pt] &= 62.64 \, \text{m/s}\end{align*}[/tex]
With the velocity determined, we now calculate the increase in temperature using the kinetic energy converted to thermal energy. The formula applied is:
[tex]$\begin{align*}\Delta T &= \frac{v^2}{2c} \\[6pt]\end{align*}[/tex]
where:
Plugging in the values:
[tex]$\begin{align*}\Delta T &= \frac{62.64^2}{2 \times 4200} \\[6pt] &= 0.47 \, ^\circ\mathrm{C}\end{align*}[/tex]
The final temperature at the bottom is obtained by adding the temperature increase to the initial temperature:
[tex]$\begin{align*}T_{\text{final}} &= T_{\text{initial}} + \Delta T \\[6pt] &= 20 \, ^\circ\mathrm{C} + 0.47 \, ^\circ\mathrm{C} \\[6pt]\therefore T_{\text{final}} &= \boxed{20.47 \, ^\circ\mathrm{C}}\end{align*}[/tex]
