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Step-by-Step Solution
Step 1: Identify the Physical Principle
The problem involves a rubber catapult stretching and projecting a stone. The key principle is
conservation of energy. The elastic potential energy stored in the stretched catapult converts
into the kinetic energy of the stone.
Step 2: Write Down the Relevant Formula
The elastic potential energy for a material under tension with Young’s modulus Y , cross-sectional
area A , original length L , and extension x is given by:
\frac{1}{2}\biggl(\frac{YA}{L}\biggr) x^2
Kinetic energy of the stone of mass m and speed v is:
\frac{1}{2} m v^2
By energy conservation:
\frac{1}{2}\biggl(\frac{YA}{L}\biggr) x^2 \;=\; \frac{1}{2} m v^2
Step 3: Substitute the Given Values
• Mass of the stone, m = 20\,\text{g} = 0.02\,\text{kg}
• Young's modulus of rubber, Y = 0.5 \times 10^9\,\text{N/m}^2
• Area of cross section, A = 10^{-6}\,\text{m}^2
• Length of catapult, L = 0.1\,\text{m}
• Extension, x = 0.04\,\text{m}
Step 4: Calculate the Left Side (Elastic Potential Energy)
Compute \bigl(\frac{YA}{L}\bigr) x^2 first:
\biggl(\frac{0.5 \times 10^9 \times 10^{-6}}{0.1}\biggr) \times (0.04)^2
= \biggl(\frac{0.5 \times 10^3}{0.1}\biggr) \times 0.0016
= \biggl(\frac{500}{0.1}\biggr) \times 0.0016 \;=\; 5000 \times 0.0016 \;=\; 8
Since the formula includes a factor of \tfrac{1}{2} :
\frac{1}{2} \times 8 = 4 \,\text{J}
Step 5: Equate to Kinetic Energy and Solve for v
The right side is \frac{1}{2} m v^2 = \frac{1}{2} \times 0.02 \times v^2 = 0.01\,v^2 .
Equating both sides:
4 \;=\; 0.01\,v^2
v^2 = \frac{4}{0.01} = 400
v = \sqrt{400} = 20\,\text{m/s}
Final Answer
The velocity of the projected stone is 20\,\text{m/s} .
