A rubber cord catapult has cross-sectional area 25mm² and initial length of rubber cord is 10cm. It is stretched to 5cm. and then released to project a missile of mass 5gm. Taking Yᵣ = 5 x 10⁸ N / m² velocity of projected missile is

The correct option is 250 ms⁻¹.

Step-by-step Explanation:

To find the velocity of the projected missile, we can apply the law of conservation of energy. The elastic potential energy stored in the stretched rubber cord is completely converted into the kinetic energy of the missile upon release.

1. Identify the given values and convert them into SI units:
Cross-sectional area of the cord, $A=25{\text{mm}}^2=25\times {10}^{-6}{\text{m}}^2$
Initial length of the cord, $L=10\text{cm}=0.1\text{m}$
Extension of the cord, $\mathrm{\Delta L}=5\text{cm}=0.05\text{m}$
Mass of the missile, $m=5\text{gm}=5\times {10}^{-3}\text{kg}$
Young's modulus of rubber, $Y_r=5\times {10}^8{\text{N/m}}^2$

2. Calculate the elastic potential energy stored in the rubber cord:
The formula for the elastic potential energy ($U$) of a stretched wire or cord is:
$U=\frac{1}{2}\times \text{Stress}\times \text{Strain}\times \text{Volume}$
Using Young's modulus, this simplifies to:
$U=\frac{1}{2}\times Y_r\times \frac{A}{L}\times {\left(\mathrm{\Delta L}\right)}^2$

Substituting the values into the formula:
$U=\frac{1}{2}\times (5\times {10}^8)\times \frac{25\times {10}^{-6}}{0.1}\times {\left(0.05\right)}^2$
$U=\frac{1}{2}\times (5\times {10}^8)\times (2.5\times {10}^{-4})\times (2.5\times {10}^{-3})$
$U=\frac{1}{2}\times 31.25\times {10}^1$
$U=156.25\text{Joules}$

3. Relate the potential energy to the kinetic energy of the missile:
By conservation of energy, the kinetic energy ($K$) of the missile is equal to the elastic potential energy stored in the cord:
$\frac{1}{2}m{v}^2=U$

Substitute the mass of the missile and solve for velocity ($v$):
$\frac{1}{2}\times (5\times {10}^{-3})\times v^2=156.25$
$2.5\times {10}^{-3}\times v^2=156.25$
$v^2=\frac{156.25}{2.5\times {10}^{-3}}$
$v^2=62500$
$v=\sqrt{62500}=250{\text{ms}}^{-1}$

Thus, the velocity of the projected missile is 250 ms⁻¹.