A cylindrical drum, open at the top, contains `30` litres of water. It drains out through a small opening at the bottom. `10` litres of water comes out in time `t_1`, the next `10`litres in a further time `t_2` and the last `10` litres in a further time `t_3` Then,

Options

A. `t_1 lt t_2 lt t_3`
B. `t_1 gt t_2 gt t_3`
C. `t_1 = t_2 =t_3`
D. `t_1 gt t_2 =t_3`

Answer

To solve the problem, we need to analyze the draining of water from a cylindrical drum through a small opening at the bottom. The key points to consider are the relationship between the height of the water in the drum and the time taken for each segment of water to drain out.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- The cylindrical drum is open at the top and contains 30 liters of water.
- Water drains out through a small opening at the bottom.

2. **Volume of Water Drained**:
- The water drains in three segments:
- First 10 liters in time \( t_1 \)
- Next 10 liters in time \( t_2 \)
- Last 10 liters in time \( t_3 \)

3. **Velocity of Efflux**:
- According to Torricelli's Law, the velocity \( v \) of efflux of water from the hole is given by:
\[ v = \sqrt{2gh} \]
where \( g \) is the acceleration due to gravity and \( h \) is the height of the water column above the hole.

4. **Effect of Water Level on Time**:
- As the water drains, the height \( h \) decreases.
- Consequently, the velocity \( v \) of the water coming out will also decrease because it is directly proportional to the square root of the height \( h \).

5. **Time Relationship**:
- The time taken to drain each segment of water is inversely related to the velocity of efflux:
\[ t \propto \frac{1}{v} \propto \frac{1}{\sqrt{h}} \]
- Therefore, as the height decreases:
- \( t_1 \) (first 10 liters) will be the least because it starts with the maximum height.
- \( t_2 \) (next 10 liters) will be greater than \( t_1 \) because the height has decreased.
- \( t_3 \) (last 10 liters) will be the greatest because the height is at its minimum.

6. **Conclusion**:
- The relationship between the times can be summarized as:
\[ t_1 < t_2 < t_3 \]
- Thus, the correct option is **Option A**.