A container of height 10 m which is open at the top. has water to its full height. Two small openings are made on the walls of the container one exactly at the middle and the other at the bottom. The ratio of the velocities with which water comes out from the middle and the bottom region respectively is

Options

A. 2
B. `1/2`
C. `sqrt2`
D. `1/sqrt2`

Explanation

To solve the problem, we need to determine the ratio of the velocities of water exiting from two openings in a container filled with water. One opening is located at the middle of the container, and the other is at the bottom.

### Step-by-Step Solution:

1. **Identify the heights of the openings**:
- The total height of the container is given as \( h = 10 \, \text{m} \).
- The height of the opening at the middle of the container is \( h_1 = \frac{h}{2} = \frac{10}{2} = 5 \, \text{m} \).
- The height of the opening at the bottom of the container is \( h_2 = h = 10 \, \text{m} \).

2. **Use Torricelli's Law**:
- According to Torricelli's Law, the velocity of fluid flowing out of an opening under the influence of gravity is given by the formula:
\[ v = \sqrt{2gh} \]
- Here, \( g \) is the acceleration due to gravity, and \( h \) is the height of the fluid above the opening.

3. **Calculate the velocities**:
- For the opening at the middle (height \( h_1 \)):
\[ v_1 = \sqrt{2g h_1} = \sqrt{2g \cdot 5} = \sqrt{10g} \]
- For the opening at the bottom (height \( h_2 \)):
\[ v_2 = \sqrt{2g h_2} = \sqrt{2g \cdot 10} = \sqrt{20g} \]

4. **Find the ratio of the velocities**:
- The ratio of the velocities \( \frac{v_1}{v_2} \) is given by:
\[ \frac{v_1}{v_2} = \frac{\sqrt{10g}}{\sqrt{20g}} = \frac{\sqrt{10}}{\sqrt{20}} = \frac{\sqrt{10}}{\sqrt{10 \cdot 2}} = \frac{1}{\sqrt{2}} \]

5. **Final Result**:
- Therefore, the ratio of the velocities with which water comes out from the middle and the bottom openings is:
\[ \frac{v_1}{v_2} = \frac{1}{\sqrt{2}} \]