The flow speeds of air on the lower and upper surfaces of the wing ofan aeroplane are v and `sqrt2v` respectively. A is the area of the wing and `rho` is the density of th e surrounding air. What is the force of the dynamic lift on the wing ?

Options

A. `1/2rhovA`
B. `1/2rhov^2A`
C. `rhov^2A`
D. `(rhov^2)/(2A)`

Explanation

To find the force of dynamic lift on the wing of an airplane, we will use Bernoulli's principle, which relates the pressure difference between the upper and lower surfaces of the wing to the velocities of the air flowing over these surfaces.

### Step-by-Step Solution:

1. **Identify the given values**:
- Speed of air on the lower surface, \( v_1 = v \)
- Speed of air on the upper surface, \( v_2 = \sqrt{2}v \)
- Area of the wing, \( A \)
- Density of air, \( \rho \)

2. **Apply Bernoulli's equation**:
Bernoulli's equation states that for incompressible flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline. For our case, we can ignore the potential energy since the height difference is not given. Thus, we can write:
\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]

3. **Rearranging Bernoulli's equation**:
Rearranging the equation to find the pressure difference \( P_1 - P_2 \):
\[ P_1 - P_2 = \frac{1}{2} \rho v_2^2 - \frac{1}{2} \rho v_1^2 \]

4. **Substituting the values of \( v_1 \) and \( v_2 \)**:
Substitute \( v_1 = v \) and \( v_2 = \sqrt{2}v \):
\[ P_1 - P_2 = \frac{1}{2} \rho (\sqrt{2}v)^2 - \frac{1}{2} \rho v^2 \]
Simplifying this gives:
\[ P_1 - P_2 = \frac{1}{2} \rho (2v^2) - \frac{1}{2} \rho v^2 = \frac{1}{2} \rho (2v^2 - v^2) = \frac{1}{2} \rho v^2 \]

5. **Calculate the lift force**:
The lift force \( F_L \) is given by the pressure difference multiplied by the area of the wing:
\[ F_L = (P_1 - P_2) \cdot A \]
Substituting the pressure difference we found:
\[ F_L = \left(\frac{1}{2} \rho v^2\right) \cdot A \]

6. **Final expression for the lift force**:
Thus, the force of dynamic lift on the wing is:
\[ F_L = \frac{1}{2} \rho v^2 A \]

### Final Answer:
The force of dynamic lift on the wing is \( F_L = \frac{1}{2} \rho v^2 A \).