$3.3.8^*.$ The balloon deforms when it weakly hits the wall, as shown in the figure. The maximum deformation of the balloon $x$ is much smaller than its radius $R$. Ignoring the change in excess pressure $\Delta p$ of air in the balloon and the elasticity of the shell, estimate the time of impact with the wall. The mass of the ball $m$.
$$y=R-x$$ $$r^2=R^2-(R-x)^2$$ $$S=\pi r^2=\pi (R^2-(R-x)^2)$$ Considering $x \ll R$ $$S=\pi r^2=\pi (R^2-(R-x)^2)=2\pi Rx$$ Newton's second law in differential form: $$-F=m\ddot{x}$$ $$m\ddot{x}+F=0 \Rightarrow m\ddot{x}+2\pi Rx \cdot \Delta p=0$$ $$\ddot{x}+\frac{2\pi R\Delta p}{m}x=0$$ Recalling the differential equation of harmonic motion $$\ddot{x}+\omega ^2x=0$$ Find the cyclic frequency $\omega$ of oscillation $$\omega=\sqrt{\frac{2\pi R\Delta p}{m}}$$ Since we're interested in half of the full period, estimated time of impact with the wall could be found as $$t=\frac{1}{2}\frac{2\pi}{\omega}=\frac{\pi}{\omega}$$ $$\boxed{t=\pi\sqrt{\frac{m}{2\pi R\Delta p}}}$$
$$t=\pi\sqrt{m/(2\pi R\Delta p)}$$