A plane is in level flight at constant speed and each of its two wings has an area of 40 `m^2`. If the speed of the air is 180 km/h over the lower wing surface and 252 km/h over the upper wing surface, the mass of the plane is ________kg. (Take air density to be 1 kg `m^(–3)` and g = 10 `ms^(–2)`)

Answer

To find the mass of the plane in level flight, we can use Bernoulli's principle to relate the pressures on the upper and lower surfaces of the wings to the lift force generated by the wings. Here’s a step-by-step solution:

### Step 1: Convert the speeds from km/h to m/s
The speeds given are:
- Speed over the lower wing surface: 180 km/h
- Speed over the upper wing surface: 252 km/h

To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \).

\[ \text{Speed of lower wing} = 180 \times \frac{5}{18} = 50 \, \text{m/s} \]
\[ \text{Speed of upper wing} = 252 \times \frac{5}{18} = 70 \, \text{m/s} \]

### Step 2: Calculate the difference in speeds
Now, we find the difference in speeds between the upper and lower wing surfaces:

\[ \Delta v = v_{\text{upper}} - v_{\text{lower}} = 70 \, \text{m/s} - 50 \, \text{m/s} = 20 \, \text{m/s} \]

### Step 3: Calculate the lift force using Bernoulli's principle
According to Bernoulli's principle, the lift force \( F \) can be expressed as:

\[ F = \frac{1}{2} \rho A (v_{\text{upper}}^2 - v_{\text{lower}}^2) \]

Where:
- \( \rho \) = density of air = 1 kg/m³
- \( A \) = total wing area = 2 wings × 40 m² = 80 m²

Calculating the lift force:

\[ F = \frac{1}{2} \times 1 \, \text{kg/m}^3 \times 80 \, \text{m}^2 \times (70^2 - 50^2) \]

Calculating \( 70^2 \) and \( 50^2 \):

\[ 70^2 = 4900 \quad \text{and} \quad 50^2 = 2500 \]
\[ 70^2 - 50^2 = 4900 - 2500 = 2400 \]

Now substituting back into the lift force equation:

\[ F = \frac{1}{2} \times 1 \times 80 \times 2400 = 96000 \, \text{N} \]

### Step 4: Relate lift force to weight of the plane
In level flight, the lift force equals the weight of the plane:

\[ F = m \cdot g \]

Where \( g = 10 \, \text{m/s}^2 \). Thus,

\[ 96000 = m \cdot 10 \]

### Step 5: Solve for the mass \( m \)
Rearranging the equation to find \( m \):

\[ m = \frac{96000}{10} = 9600 \, \text{kg} \]

### Final Answer
The mass of the plane is **9600 kg**.

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