Question

A rubber cord has a cross-sectional area $1{\text{mm}}^2$ and total unstretched length $10.0\text{cm}$. It is stretched to $12.0\text{cm}$ and then released to project a missile of mass $5.0\text{g}$. Taking Young's modulus $Y$ for rubber as $5.0\times {10}^8{\text{N/m}}^2$. The velocity of projection is _______${\text{ms}}^{-1}$.

Answer 1

Equivalent force constant of rubber cord,

$k=\frac{YA}{l}=\frac{(5.0\times {10}^8)(1.0\times {10}^{-6})}{\left(0.1\right)}=5\times {10}^3\text{N/m}$

Now, from conservation of mechanical energy,

Elastic potential energy of cord $=$ Kinetic energy of missile

$\Rightarrow \frac{1}{2}k(\Delta l{)}^2=\frac{1}{2}m{v}^2$

$\Rightarrow v=\left(\sqrt{\frac{k}{m}}\right)\Delta l$

$\Rightarrow v=\left(\sqrt{\frac{5\times {10}^3}{5\times {10}^{-3}}}\right)(12.0-10.0)\times {10}^{-2}$

$\Rightarrow v=20\text{m/s}$

Note:
The two assumptions made in the question are -

$1)$. $k$ has been assumed to be constant.

$2)$. The entire potential energy is getting converted to kinetic energy of the missile.