A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section `10^(-6)m^(2)` stretched by an amount 0.04 m. The velocity of the projected stone is __________ m/s.
(Young's modulus of rubber `=0.5 xx 10^(9) N//m^(2)`)

Answer

To solve the problem, we need to find the velocity of a stone projected from a rubber catapult using the principles of energy conservation and Young's modulus.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mass of the stone, \( m = 20 \, \text{g} = 0.02 \, \text{kg} \) (convert grams to kilograms)
- Length of the rubber catapult, \( L = 0.1 \, \text{m} \)
- Area of cross-section, \( A = 10^{-6} \, \text{m}^2 \)
- Stretch of the rubber, \( x = 0.04 \, \text{m} \)
- Young's modulus of rubber, \( Y = 0.5 \times 10^9 \, \text{N/m}^2 \)

2. **Calculate the Spring Constant (k):**
The spring constant \( k \) for the rubber can be calculated using the formula:
\[ k = \frac{Y \cdot A}{L} \]
Substituting the values:
\[ k = \frac{(0.5 \times 10^9) \cdot (10^{-6})}{0.1} = \frac{0.5 \times 10^3}{0.1} = 5 \times 10^3 \, \text{N/m} \]

3. **Calculate the Potential Energy Stored in the Stretched Rubber:**
The potential energy \( PE \) stored in the rubber when stretched is given by:
\[ PE = \frac{1}{2} k x^2 \]
Substituting the values:
\[ PE = \frac{1}{2} \cdot (5 \times 10^3) \cdot (0.04)^2 = \frac{1}{2} \cdot (5 \times 10^3) \cdot (0.0016) = 4 \, \text{J} \]

4. **Use Energy Conservation to Find the Velocity:**
According to the conservation of energy, the potential energy stored in the rubber will be converted into kinetic energy \( KE \) of the stone when it is projected:
\[ PE = KE \]
The kinetic energy is given by:
\[ KE = \frac{1}{2} m v^2 \]
Setting the two energies equal:
\[ 4 = \frac{1}{2} \cdot (0.02) \cdot v^2 \]

5. **Solve for Velocity (v):**
Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{4 \cdot 2}{0.02} = \frac{8}{0.02} = 400 \]
Taking the square root:
\[ v = \sqrt{400} = 20 \, \text{m/s} \]

### Final Answer:
The velocity of the projected stone is \( \boxed{20} \, \text{m/s} \).