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Gravitational potential energy (GPE) is the energy an object has due to its position in a gravitational field. When water is stored at a higher elevation, such as in a reservoir or behind a dam, it possesses potential energy because of its height above ground level. This stored energy can be released when the water is allowed to flow downwards.
In mathematical terms, the gravitational potential energy of an object is calculated using the formula:

• \[ U = mgh \]

where:

• \( m \) is the mass of the object (or water in kilograms),
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \) on Earth), and
• \( h \) is the height of the object above a reference point (usually the ground or the lowest point in a system).

In our case, the water at the top of a 100-meter waterfall has gravitational potential energy because of its height. As it falls, this energy converts into kinetic energy, which can then be harnessed for doing work, such as generating electricity in hydroelectric power stations.

Energy conversion is a critical process in systems like hydroelectric power plants. It involves changing one form of energy into another. In the context of hydroelectric power, we primarily deal with the conversion of gravitational potential energy into kinetic energy, and then further into electrical energy.
When water falls from a height, its potential energy due to gravity is transformed into kinetic energy. This is because energy cannot be created or destroyed, only converted from one form to another (according to the law of conservation of energy). This conversion happens as water accelerates downward through the force of gravity, increasing its speed and kinetic energy.
Most hydroelectric plants use turbines, which are turned by the flowing water. As the turbine blades spin, they drive a generator that converts kinetic energy into electrical energy, which can then be distributed and used to power various applications. In our example, three-fourths of this kinetic energy is successfully converted into usable electrical energy, indicating the efficiency of the system in capturing and transforming energy resources.

The mass flow rate is a crucial concept in fluid dynamics and relates to the amount of mass moving through a system per unit of time. In hydroelectric plants, it refers to the mass of water passing through the system every second. This measurement is essential for determining how much energy can be generated from a hydroelectric plant, as it directly affects both the potential and kinetic energies involved.
For our waterfall problem, given a volume flow rate of 1200 cubic meters per second and knowing the density of water is 1000 kilograms per cubic meter, the mass flow rate \( \dot{m} \) can be calculated using the formula:

• \[ \dot{m} = \text{Volume flow rate} \times \text{Density} \]

Plugging in the values, we find:

• \[ \dot{m} = 1200 \, \text{m}^3/ \text{s} \times 1000 \, \text{kg/m}^3 = 1,200,000 \, \text{kg/s} \]

This mass flow rate tells us how much water, in terms of mass, is flowing past a point every second, allowing us to calculate the potential energy available for conversion into electricity.