The height of Niagra falls is 50 m. If J = 4.2 J/cal, the difference in temperature of water at top level and at the bottom level of fall will be :

Question

Accepted Answer

To determine the difference in temperature of water at the top and bottom of Niagara Falls, we can use the principle of conservation of energy. When water falls from a height, its potential energy is converted into kinetic energy and then into thermal energy, which causes a slight increase in temperature.

1. **Calculation of Potential Energy:**
   - The potential energy (PE) of water at the top is given by the formula: 
     $$ PE = m \cdot g \cdot h $$
     where $ m $ is the mass of water, $ g $ is the acceleration due to gravity (approximately $ 9.8 \, \text{m/s}^2 $), and $ h $ is the height (50 m).

2. **Conversion to Thermal Energy:**
   - The potential energy is converted into thermal energy, which increases the temperature of the water. The increase in thermal energy can be expressed as:
     $$ \Delta Q = m \cdot c \cdot \Delta T $$
     where $ c $ is the specific heat capacity of water (approximately $ 1 \, \text{cal/g°C} $) and $ \Delta T $ is the change in temperature.

3. **Equating Potential Energy to Thermal Energy:**
   - Since the potential energy is converted into thermal energy, we have:
     $$ m \cdot g \cdot h = m \cdot c \cdot \Delta T $$
   - Simplifying, we get:
     $$ g \cdot h = c \cdot \Delta T $$

4. **Substituting Values:**
   - Substituting the known values:
     $$ 9.8 \, \text{m/s}^2 \cdot 50 \, \text{m} = 1 \, \text{cal/g°C} \cdot \Delta T $$
   - Converting $ 9.8 \, \text{m/s}^2 $ to calories using the conversion factor $ 1 \, \text{J} = 4.2 \, \text{cal} $:
     $$ \Delta T = \frac{9.8 \cdot 50}{4.2} $$

5. **Calculating the Temperature Change:**
   - $$ \Delta T = \frac{490}{4.2} \approx 0.117 \, \text{°C} $$

Therefore, the difference in temperature of water at the top and bottom of Niagara Falls is approximately $ 0.117 \, \text{°C} $, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (0.49°C):** This value is too high. The calculation shows that the temperature change is much smaller due to the specific conversion factor and the height involved.
- **Option 3 (0.82°C):** This value is also too high. The energy conversion from potential to thermal energy does not support such a large temperature increase.
- **Option 4 (2.05°C):** This value is significantly higher than the calculated result. The physics of energy conversion at this height does not allow for such a large temperature change.

In conclusion, the correct answer is Option 1, as the calculated temperature difference is $ 0.117 \, \text{°C} $.