A boy's catapult is made of rubber cord which is $42\ \text{cm}$ long, with $6\ \text{mm}$ diameter of cross-section and of negligible mass. The boy keeps a stone weighing $0.02\ \text{kg}$ on it and stretches the cord by $20\ \text{cm}$ by applying a constant force. When released, the stone flies off with a velocity of $20\ \text{m s}^{-1}$. Neglect the change in the area of cross-section of the cord while stretched. The Young's modulus of rubber is closest to
Answer: (A) $10^6\ \text{N m}^{-2}$
The elastic energy stored in the cord becomes the stone's kinetic energy:
$$\frac12\frac{YA}{L}x^2 = \frac12mv^2 = \frac12\times0.02\times400 = 4\ \text{J}$$
With $A = \pi(3\times10^{-3})^2 = 2.83\times10^{-5}\ \text{m}^2$, $L = 0.42\ \text{m}$, $x = 0.2\ \text{m}$:
$$Y = \frac{8L}{Ax^2} = \frac{8\times0.42}{2.83\times10^{-5}\times0.04} \approx 3\times10^6\ \text{N m}^{-2}$$
This is of the order of $10^6\ \text{N m}^{-2}$.
Solution by Sreeraj P, M.Sc Physics