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 find the initial velocity of efflux of the liquid from the hole in the vessel, we can use Bernoulli's equation. Here’s a step-by-step solution:

### Step 1: Identify the heights
The total height of the liquid in the vessel is given as 15 m. The hole is made at a height of 10 m from the base. Therefore, the height of the liquid column above the hole is:
\[ h = 15 \, \text{m} - 10 \, \text{m} = 5 \, \text{m} \]

### Step 2: Apply Bernoulli's equation
According to Bernoulli's principle, the sum of pressure energy, potential energy, and kinetic energy per unit volume is constant. We can apply Bernoulli's equation between two points:
- Point 1: At the surface of the liquid (where the velocity is negligible)
- Point 2: At the hole (where the liquid is exiting)

The equation can be written as:
\[ P_1 + \rho g h_1 + \frac{1}{2} \rho v_1^2 = P_2 + \rho g h_2 + \frac{1}{2} \rho v_2^2 \]

### Step 3: Assign values to the equation
At point 1 (surface of the liquid):
- Pressure \( P_1 = P_0 \) (atmospheric pressure)
- Height \( h_1 = 5 \, \text{m} \) (height of liquid above the hole)
- Velocity \( v_1 = 0 \) (negligible)

At point 2 (the hole):
- Pressure \( P_2 = P_0 \) (atmospheric pressure)
- Height \( h_2 = 0 \) (reference level at the hole)
- Velocity \( v_2 = v_e \) (efflux velocity)

Substituting these values into Bernoulli's equation gives:
\[ P_0 + \rho g (5) + 0 = P_0 + 0 + \frac{1}{2} \rho v_e^2 \]

### Step 4: Simplify the equation
Canceling \( P_0 \) from both sides:
\[ \rho g (5) = \frac{1}{2} \rho v_e^2 \]

Dividing both sides by \( \rho \):
\[ g (5) = \frac{1}{2} v_e^2 \]

### Step 5: Solve for \( v_e \)
Rearranging the equation gives:
\[ v_e^2 = 2g(5) \]
\[ v_e = \sqrt{2g(5)} \]

### Step 6: Substitute the value of \( g \)
Given \( g = 10 \, \text{m/s}^2 \):
\[ v_e = \sqrt{2 \times 10 \times 5} \]
\[ v_e = \sqrt{100} \]
\[ v_e = 10 \, \text{m/s} \]

### Final Answer
The initial velocity of efflux is:
\[ \boxed{10 \, \text{m/s}} \]

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