Question

A river descends \(15 \mathrm{~m}\) in passing through rapids. The speed of the water is \(3.2 \mathrm{~m} / \mathrm{s}\) upon entering the rapids and is \(13 \mathrm{~m} / \mathrm{s}\) as it leaves. What percentage of the potential energy lost by the water in traversing the rapids appears as kinetic energy of water downstream? What happens to the rest of the energy?

Short answer

The percentage of potential energy lost by the water in traversing the rapids that appears as kinetic energy downstream can be calculated using the aforementioned steps. Any remaining energy has likely been converted into other forms, such as heat or sound energy due to the turbulence of the rapids.

Step-by-step solution

Calculating Initial and Final Potential Energy

Potential energy (PE) is given by the formula: PE = mgh. Here, m is the mass of the water, g is acceleration due to gravity (approx. 9.81 m/s²), and h is the height of the descent (\(15 \mathrm{~m}\)). The mass cancels out in this problem as we are asked for the percentage of potential energy converted, not the exact energy amount. Thus, we will find the initial potential energy as \(PE_{initial} = m \cdot g \cdot h_{initial}\) and the final potential energy as \(PE_{final} = m \cdot g \cdot h_{final}\). Considering that the water descends from the rapids, the final height is 0, hence \(PE_{final} = 0\).

Calculating Initial and Final Kinetic Energy

Kinetic energy (KE) is determined by the formula: KE = 1/2 m v². Like in the previous step, the mass will cancel out as we are looking for the percentage change. Hence, we can calculate initial kinetic energy as \(KE_{initial} = 1/2 \cdot m \cdot v_{initial}^2\) and the final kinetic energy as \(KE_{final} = 1/2 \cdot m \cdot v_{final}^2\).

Calculating Energy Change

The potential energy lost by the water is \(PE_{initial} - PE_{final}\) and the kinetic energy it gains is \(KE_{final} - KE_{initial}\). The percentage of potential energy lost that appears as kinetic energy is then \((KE_{final} - KE_{initial}) / (PE_{initial} - PE_{final}) \cdot 100%\).

Interpreting the Remaining Energy

Energy can neither be created nor destroyed (Law of Energy Conservation). Therefore, any energy not converted from potential to kinetic must have been dissipated in other forms, such as heat or sound due to the turbulence in the rapids.

Key concepts

Potential Energy

Potential energy is the energy stored within an object due to its position relative to some zero position. An object possesses gravitational potential energy if it is positioned at a height above the ground. To figure out this type of energy, we use the formula:
\[ PE = m \times g \times h \]
where \( m \) represents the object's mass, \( g \) the acceleration due to gravity (on Earth approximate to \( 9.81 \text{ m/s}^2 \)), and \( h \) is the height from the zero position.
In fluid dynamics, this concept explains how water at the top of rapids has stored energy that can be converted into other forms of energy as it flows down. The energy conversion begins when this potential energy starts to change into kinetic energy, with the water's mass moving downhill due to gravity.

Kinetic Energy

Kinetic energy is the energy of motion. It is given to an object when work is done upon it, causing it to move. The kinetic energy (KE) of an object can be calculated using the equation:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity. In the context of fluid dynamics, when water in a river flows from a higher elevation and accelerates through rapids, its velocity increases, and thereby its kinetic energy increases. This transition from slow-moving water to fast-moving water illustrates kinetic energy at work.
Understanding this energy increase helps to quantify the amount of potential energy that is being converted as the water speeds up.

Energy Conversion in Fluids

The energy conversion in fluids is a fundamental principle where the energy within a fluid changes from one form to another. In the case of a river flowing through rapids:

• Gravitational potential energy is converted to kinetic energy as the river descends.
• The energy lost from the potential energy store is gained in the kinetic energy of the moving water.

It is essential to note that some energy is inevitably lost to the surroundings due to turbulence, friction, and other resistive forces, converting to thermal energy or sound. This conversion plays a critical role in understanding how energy is redistributed and utilized in fluid systems.

Mechanical Energy Conservation

The principle of mechanical energy conservation states that in the absence of non-conservative forces (such as friction), the total mechanical energy of a system remains constant. It embodies one of the fundamental laws of physics – the law of conservation of energy – which asserts that energy cannot be created or destroyed, only transformed from one form to another.
When discussing fluids, like in our river rapids example, mechanical energy conservation allows us to understand that the decrease in potential energy as the water falls must equal the increase in kinetic energy plus any energy lost to the environment (for instance, as heat due to friction with the riverbed).

• Calculating the percentage of potential energy that appears as kinetic can give insights into the efficiency of this energy conversion process.
• The energy not observed in the kinetic form is accounted for by identifying alternative conversion paths, such as heat, sound, or other forms of energy dissipation.

Therefore, interpreting energy changes in fluid dynamics requires the assessment of both visible mechanical energy changes and more subtle forms of energy transfer.