An open vessel containing the liquid upto a height of 15 m. A small hole is made at height of 10 m from the base of the vessel then the initial velocity of efflux is (g = 10 m/`s^2`)

Options

A. 1 m/s
B. `10sqrt(2) m/s`
C. 5 m/s
D. 10 m/s

Answer

To solve the problem of finding the initial velocity of efflux from a hole in an open vessel, we can use Torricelli's theorem. The theorem states that the speed of efflux of a fluid under the force of gravity through an opening is given by the equation:

\[ v = \sqrt{2gh} \]

Where:
- \( v \) is the velocity of efflux,
- \( g \) is the acceleration due to gravity,
- \( h \) is the height of the liquid column above the hole.

### Step-by-step Solution:

1. **Identify the height of the liquid column above the hole:**
- The total height of the liquid in the vessel is 15 m.
- The height of the hole from the base of the vessel is 10 m.
- Therefore, the height of the liquid column above the hole is:
\[ h = \text{Total height} - \text{Height of the hole} = 15 \, \text{m} - 10 \, \text{m} = 5 \, \text{m} \]

2. **Substitute the values into the formula:**
- We know \( g = 10 \, \text{m/s}^2 \) and \( h = 5 \, \text{m} \).
- Now, we can substitute these values into the equation for velocity:
\[ v = \sqrt{2gh} = \sqrt{2 \times 10 \, \text{m/s}^2 \times 5 \, \text{m}} \]

3. **Calculate the value inside the square root:**
- First, calculate \( 2 \times 10 \times 5 \):
\[ 2 \times 10 \times 5 = 100 \]

4. **Take the square root:**
- Now, we find the square root of 100:
\[ v = \sqrt{100} = 10 \, \text{m/s} \]

5. **Final answer:**
- The initial velocity of efflux is:
\[ v = 10 \, \text{m/s} \]

### Summary:
The initial velocity of efflux from the hole in the vessel is \( 10 \, \text{m/s} \).