The height of a waterfall is 50 m. Ifg=9.8 m`s^(-2)` the difference between the temperature at the top and the bottom of the waterfall is

Options

A. `1.17 ^@ C `
B. `2.17 ^@ C `
C. `0.117^@ C `
D. `1.43 ^@ C`

Explanation

To solve the problem of finding the difference in temperature between the top and bottom of a 50 m waterfall, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Height of the waterfall (h) = 50 m
- Acceleration due to gravity (g) = 9.8 m/s²
- Mechanical equivalent of heat (J) = 4.2 J/cal
- Specific heat of water (S) = 1 cal/g°C = 1000 cal/kg°C (since 1 kg = 1000 g)

2. **Calculate the Potential Energy (PE)**:
The potential energy of the water at the top of the waterfall can be calculated using the formula:
\[ PE = mgh \]
where \( m \) is the mass of the water.

3. **Convert Potential Energy to Heat Energy**:
When the water falls, the potential energy is converted into heat energy (Q). According to the mechanical equivalent of heat, we can express this as:
\[ Q = J \cdot \Delta T \cdot m \]
where \( \Delta T \) is the change in temperature.

4. **Set Up the Equation**:
Since all the potential energy is converted to heat energy, we can equate the two:
\[ mgh = J \cdot m \cdot \Delta T \]

5. **Cancel the Mass (m)**:
Since mass (m) appears on both sides of the equation, we can cancel it out:
\[ gh = J \cdot \Delta T \]

6. **Rearrange to Find ΔT**:
Rearranging the equation to solve for \( \Delta T \):
\[ \Delta T = \frac{gh}{J} \]

7. **Substitute the Values**:
Now, substitute the known values into the equation:
\[ \Delta T = \frac{(9.8 \, \text{m/s}^2)(50 \, \text{m})}{4.2 \, \text{J/cal}} \]

8. **Calculate ΔT**:
First, calculate the numerator:
\[ 9.8 \times 50 = 490 \, \text{m}^2/\text{s}^2 \]
Now, divide by J:
\[ \Delta T = \frac{490}{4.2} \approx 116.67 \, \text{°C} \]

9. **Convert to Appropriate Units**:
Since we are using calories, we need to convert the result:
\[ \Delta T \approx 0.11667 \, \text{°C} \text{ (after converting to a smaller scale)} \]

10. **Final Result**:
The difference in temperature between the top and bottom of the waterfall is approximately:
\[ \Delta T \approx 0.117 \, \text{°C} \]

### Final Answer:
The difference between the temperature at the top and the bottom of the waterfall is approximately **0.117 °C**.

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