91Ó°ÊÓ

Potential energy is the stored energy of an object due to its position or height. In the context of the waterfall exercise, we focus on the potential energy since the water is elevated at a significant height. Potential energy can be calculated using the formula:\[ PE = mgh \]Here:

• \( m \) is the mass of the object (in this case, the waterfall's water),
• \( g \) is the acceleration due to gravity, which is approximately \( 9.8 \, m/s^2 \), and
• \( h \) is the height from which the water falls, measured in meters.

For the waterfall problem, since the task was simplified by considering the mass of water as \( 1 \, kg \), we can directly calculate the potential energy from the product of gravity and height:\[ PE = 9.8 \, m/s^2 \times 807 \, m = 7908.6 \, J \]This is the potential energy converted into kinetic and, eventually, thermal energy as the water descends from the top to the bottom of the falls.

Thermal energy refers to the internal energy present in a system due to the random motion of its particles. In this problem, as the water falls, its potential energy is progressively transformed into thermal energy. This transformation occurs because the falling water's kinetic energy, once at the bottom, is assumed to convert into heat, raising the temperature of the water.When the potential energy calculated (\[ 7908.6 \, J \]) is entirely transferred as heat (\( Q \)) to the water, it allows us to understand how much energy is used to increase the water's temperature. In this example, the transfer of potential energy to thermal energy is a perfect scenario assuming no energy is lost to the environment. This ideal conversion helps illustrate the impact gravity and height can have on temperature changes in a closed system.

The change in temperature, denoted as \( \Delta T \), represents the difference between the final temperature and the initial temperature in a thermodynamic system. To determine how much the temperature of water at the base of Angel Falls increases, we employ the concept where the heat gained by water equals the change in its temperature.We use the formula:\[ Q = mc\Delta T \]By rearranging this equation for \( \Delta T \), we have:\[ \Delta T = \frac{Q}{mc} \]Where:

• \( Q \) is the thermal energy or heat, equivalent to the calculated potential energy \( 7908.6 \, J \),
• \( m \) is the mass (considered as \( 1 \, kg \)), and
• \( c \) is the specific heat capacity of water, which is approximately \( 4180 \, J/kg^{\circ}C \).

Substituting these values, the change in temperature is computed as:\[ \Delta T = \frac{7908.6}{4180} \approx 1.89^{\circ}C \]The final temperature of the water at the waterfall's base is the initial temperature plus this change:\[ T_{final} = 15.0^{\circ}C + 1.89^{\circ}C \approx 16.89^{\circ}C \]Thus, the maximum temperature increase from top to bottom is approximately \( 1.89^{\circ}C \), reflecting the impact of gravitational potential energy conversion into thermal energy.