Question

A rubber cord of a catapult has a cross sectional area $1{\text{mm}}^2$ and total unstretched length $10\text{cm}$. It is stretched to $12\text{cm}$ and then released to project a stone of mass $5\text{gm}$. Taking Young's Modulus $Y$ for rubber as $5\times {10}^8{\text{N/m}}^2$, the velocity of projection is

• A. $10\text{m/s}$
• B. $15\text{m/s}$
• C. $20\text{m/s}$
• D. $25\text{m/s}$

Answer 1

The correct option is C $20\text{m/s}$

The potential energy of a rubber cord catapult is given by

$U=\frac{1}{2}\frac{YA{l}^2}{L}$

Here, $A$ is the cross sectional area, $Y$ is the Young's Modulus for rubber, $L$ is the original length and $l$ is the change in length.

Now, the kinetic energy of the mass is given by,

$KE=\frac{1}{2}m{v}^2$

where, $m$ is the mass of the projectile and $v$ is the velocity

From energy conservation we know that,

$U=KE$

$\frac{1}{2}m{v}^2=\frac{1}{2}\frac{YA{l}^2}{L}$

$v=\left(\sqrt{\frac{YA}{mL}}\right)l$

Given that,

$Y=5\times {10}^8{\text{N/m}}^2$

$A=1{\text{mm}}^2={10}^{-6}{\text{m}}^2$

$m=5\text{gm}=5\times {10}^{-3}\text{Kg}$

$L=10\text{cm}=10\times {10}^{-2}\text{m}$

$l=12-10=2\text{cm}=2\times {10}^{-2}\text{m}$

Putting in all the given values,

$v=\left(\sqrt{\frac{5\times {10}^8\times {10}^{-6}}{5\times {10}^{-3}\times 10\times {10}^{-2}}}\right)\times 2\times {10}^{-2}$

$v=2\times {10}^3\times {10}^{-2}$

$v=20{\text{ms}}^{-1}$