A hole is made at the bottom of the tank filled with water (density `=1000 kgm^(-3))`. If the total pressure at the bottom of the tank is three atmospheres (1 atmosphere `=10^(5) Nm^(-2))`, then the velocity of efflux is nearest to

Options

A. `sqrt(400) ms^(-1)`
B. `sqrt(200) ms^(-1)`
C. `sqrt(600) ms^(-1)`
D. `sqrt(500) ms^(-1)`

Explanation

To solve the problem of finding the velocity of efflux from a hole at the bottom of a tank filled with water, we can use Bernoulli's principle. Here's a step-by-step solution:

### Step 1: Understand the Given Information
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \)
- Total pressure at the bottom of the tank, \( P = 3 \, \text{atmospheres} \)
- Conversion of atmospheric pressure: \( 1 \, \text{atmosphere} = 10^5 \, \text{N/m}^2 \)

### Step 2: Calculate Total Pressure in Pascals
Convert the total pressure from atmospheres to Pascals:
\[ P = 3 \, \text{atmospheres} = 3 \times 10^5 \, \text{N/m}^2 = 300000 \, \text{N/m}^2 \]

### Step 3: Apply Bernoulli’s Equation
According to Bernoulli's principle, the pressure energy per unit volume is converted to kinetic energy per unit volume. The equation can be expressed as:
\[ \Delta P = \frac{1}{2} \rho v^2 \]
Where \( \Delta P \) is the pressure difference and \( v \) is the velocity of efflux.

### Step 4: Determine the Pressure Difference
Since the pressure at the surface of the water is atmospheric pressure (1 atmosphere), the pressure difference \( \Delta P \) at the hole is:
\[ \Delta P = P - P_{\text{atmospheric}} = 3 \times 10^5 \, \text{N/m}^2 - 1 \times 10^5 \, \text{N/m}^2 = 2 \times 10^5 \, \text{N/m}^2 \]

### Step 5: Substitute Values into the Equation
Now, substitute \( \Delta P \) into Bernoulli's equation:
\[ 2 \times 10^5 = \frac{1}{2} \times 1000 \times v^2 \]

### Step 6: Solve for Velocity \( v \)
Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{2 \times 2 \times 10^5}{1000} \]
\[ v^2 = \frac{4 \times 10^5}{1000} = 400 \]
Taking the square root:
\[ v = \sqrt{400} = 20 \, \text{m/s} \]

### Final Answer
The velocity of efflux is nearest to:
\[ \boxed{20 \, \text{m/s}} \]