$2.3.9.$ A slingshot was made from a long rubber band of stiffness $k$. Find the kinetic energy of a "projectile" fired from this slingshot if the rubber band was stretched with a force $F$ and then released.
Solution
When the slingshot is stretched by a distance $x$ from both ends, the total pulling force $F$ is balanced by the tension in the two halves of the rubber band. Let's write the equilibrium condition: $$\frac{F}{2} = kx$$ Expressing the elongation $x$ of the rubber band: $$x = \frac{F}{2k} \quad (a)$$
The potential energy of the stretched rubber band at this point is: $$E_{\text{pot}} = \frac{kx^2}{2}$$ Substituting expression (a) into the energy formula: $$E_{\text{pot}} = \frac{kF^2}{8k^2} = \frac{F^2}{8k}$$
From the law of conservation of energy, all the potential energy of the elastic band $E_{\text{pot}}$ converts into the kinetic energy of the projectile $E_k$: $$E_{\text{pot}} = E_k \implies E_k = \frac{F^2}{8k}$$