Problem 30 - College Physics

Question

Much of the energy of falling water in a waterfall is converted into heat. If all the mechanical energy is converted into heat that stays in the water, how much of a rise in temperature occurs in a \(100 \mathrm{~m}\) waterfall?

Step-by-Step Solution

Answer

The temperature of the water rises by approximately 0.23 °C.

Step 1: Understand the Energy Conversion

When water falls from a height, its potential energy is converted into kinetic energy, and finally, into heat energy, which increases the temperature of the water. To find the temperature rise, we need to calculate the potential energy lost by the water.

Step 2: Determine the Potential Energy Lost

The potential energy lost by the water is given by the formula \( PE = mgh \), where \( m \) is the mass of the water, \( g \) is the acceleration due to gravity (approximately \( 9.81 \mathrm{~m/s^2} \)), and \( h \) is the height of the fall which is \( 100 \mathrm{~m} \).

Step 3: Relate Potential Energy to Heat Energy

The potential energy lost is converted into heat energy. The heat energy gained by the water is \( Q = mc \Delta T \), where \( c \) is the specific heat capacity of water (approximately \( 4184 \mathrm{~J/kg \, °C} \)) and \( \Delta T \) is the change in temperature.

Step 4: Equate and Solve for Temperature Change

Since all potential energy is converted to heat, equate the two equations: \( mgh = mc \Delta T \). Simplify to find \( \Delta T \):\[ \Delta T = \frac{gh}{c} \]Substitute the values: \( g = 9.81 \mathrm{~m/s^2} \), \( h = 100 \mathrm{~m} \), \( c = 4184 \mathrm{~J/kg \, °C} \).

Step 5: Calculate \( \Delta T \)

Calculate the change in temperature:\[ \Delta T = \frac{9.81 \times 100}{4184} \approx 0.234 \mathrm{~°C} \]

Step 6: Conclude with the Temperature Rise

The rise in temperature of the water from a 100-meter waterfall, converting all mechanical energy into heat, is approximately \( 0.23 \mathrm{~°C} \).

Key Concepts

Potential Energy

Potential energy is the stored energy of an object due to its position or height. In the context of a waterfall, potential energy refers to the energy possessed by the water because of its elevated position above the ground.
When water is at the top of the waterfall, it is full of potential energy due to gravity acting on its mass. This gravitational potential energy can be calculated using the formula:

• \( PE = mgh \)
- Here, \( m \) is the mass of the water, \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)), and \( h \) is the height of the fall.

As the water begins to fall, this potential energy is converted into other forms of energy, showcasing how energy can shift from one form to another in nature.

Kinetic Energy

Kinetic energy is the energy of motion. As water cascades down a waterfall, its potential energy transforms into kinetic energy.
The water picks up speed as it falls, showcasing how energy doesn't disappear but transfers from one type to another. This change can be observed vividly in the roaring movement of water as it reaches the bottom of a waterfall. The energy associated with this motion can be calculated using its velocity by the formula:

• \( KE = \frac{1}{2}mv^2 \)
- Here, \( KE \) is kinetic energy, \( m \) is mass, and \( v \) is velocity.

This formula helps illustrate the fascinating interplay between potential and kinetic energy during the waterfall's descent. Ultimately, much of this kinetic energy is converted into heat energy upon impact.

Heat Energy

Heat energy is what results from the conversion of mechanical energy during the waterfall's journey. Upon reaching the bottom, the kinetic energy due to the water's motion translates into heat energy.
This process of energy conversion is crucial to understanding how temperature changes in the water occur as a result of the fall.

• The heat energy gained by the water can be expressed as \( Q = mc\Delta T \), where:
  • \( Q \) represents the heat energy.
  • \( m \) is the mass of the water.
  • \( c \) is the specific heat capacity of water.
  • \( \Delta T \) is the change in temperature.

All the potential energy that began at the top of the waterfall transforms into heat energy by the end of the fall, influencing the water's temperature.

Specific Heat Capacity

Specific heat capacity is a property that indicates the amount of heat energy required to change the temperature of a unit mass of a substance by one degree Celsius. For water, this value is fairly high, meaning it takes a considerable amount of energy to change its temperature.
The specific heat capacity of water is approximately \( 4184 \, \text{J/kg°C} \). This feature is crucial in understanding how even the massive energy from a waterfall results only in a slight rise in water temperature.

• In the equation \( Q = mc\Delta T \), the specific heat capacity \( c \) is a fixed value for water, allowing us to calculate \( \Delta T \), the temperature change.

Understanding this concept highlights the resilience of water’s temperature, even when infused with significant energy, like in the case of a waterfall.