# A boy's catapult is made of rubber cord which is $42\; cm$ long, with $6\; mm$ diameter of cross-section and of negligible mass. The boy keeps a stone weighing $0.02\; kg$ on it and stretches the cord by $20\; cm$ by applying a constant force. When released, the stone flies off with a velocity of $20\; ms^{-1}$. Neglect the change in the area of cross-section of the cord while stretched. The young's modulus of rubber is closest to :  Option 1)$10^{8}\; N\! m^{-2}$Option 2)$10^{3}\; N\! m^{-2}$Option 3)$10^{6}\; N\! m^{-2}$  Option 4)$10^{4}\; N\! m^{-2}$

For string

$\\l=42\; cm\\d=6\; mm\Rightarrow r=3\; mm\\A=\pi\; r^{2}\\\Delta l=20\; cm$

for stone

$\\m=0.02\; kg\\V=20\; m/s$

So apply energy conservation.

P.E. stored in string = K.E of stone

$\Rightarrow \frac{1}{2}\times Y\times \left ( \frac{\Delta L}{L} \right )^{2}\times A\times L=\frac{1}{2}\; mv^{2}$

$\Rightarrow Y = \frac{mv^{2}\times L}{(\Delta L)^{2}\times A}$

$= \frac{2\times 10^{-2}\times 20\times 20\times 42\times 10^{-2}}{20\times 20\times 10^{-4}\times 3.14 \times 9\times 10^{-6} }$

$Y\approx 10^{6}\; N/m^{2}$

Option 1)

$10^{8}\; N\! m^{-2}$

Option 2)

$10^{3}\; N\! m^{-2}$

Option 3)

$10^{6}\; N\! m^{-2}$

Option 4)

$10^{4}\; N\! m^{-2}$

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