The height of a waterfall is 84 metre . Assuming that the entire kinetic energy of falling water is converted into heat, the rise in temperature of the water will be (`g=9.8 m//s^(2),J=4.2` joule / cal)

Options

A. `0.196^(@)C`
B. `1.960^(@)C`
C. `0.96^(@)C`
D. `0.0196^(@)C`

Answer

To find the rise in temperature of the water when it falls from a height of 84 meters, we can use the principle of conservation of energy. The potential energy lost by the water as it falls is converted into heat energy, which raises the temperature of the water.

### Step-by-Step Solution:

1. **Identify the potential energy lost**:
The potential energy (PE) lost by the water when it falls from a height \( h \) is given by the formula:
\[ PE = mgh \]
where:
- \( m \) is the mass of the water (which will cancel out),
- \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity,
- \( h = 84 \, \text{m} \) is the height of the waterfall.

2. **Calculate the potential energy**:
Substituting the values into the potential energy formula:
\[ PE = m \cdot 9.8 \cdot 84 \]
This simplifies to:
\[ PE = 823.2m \, \text{Joules} \]

3. **Relate potential energy to heat energy**:
The heat energy gained by the water (Q) is equal to the potential energy lost:
\[ Q = m \cdot c \cdot \Delta T \]
where:
- \( c = 4.2 \, \text{J/cal} \) is the specific heat capacity of water,
- \( \Delta T \) is the rise in temperature.

4. **Set the equations equal**:
Since all potential energy is converted to heat energy:
\[ 823.2m = m \cdot 4.2 \cdot \Delta T \]
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ 823.2 = 4.2 \cdot \Delta T \]

5. **Solve for \( \Delta T \)**:
Rearranging the equation to find \( \Delta T \):
\[ \Delta T = \frac{823.2}{4.2} \]
Calculating the value:
\[ \Delta T \approx 196 \, \text{degrees Celsius} \]

6. **Convert to appropriate units**:
Since the question asks for the rise in temperature in degrees Celsius, we can express the result directly as:
\[ \Delta T \approx 196 \, \text{°C} \]

### Final Answer:
The rise in temperature of the water will be approximately **196 °C**.