$3.1.6.$ Determine the maximum velocity of a ball of a mathematical pendulum of length $l$ moving in one plane if the amplitude of the displacement at small oscillations of the pendulum is equal to $x_0$.
The motion of a mathematical pendulum is described by the solution of the equation of harmonic oscillations $$ x = x_0\sin\omega t $$ Where $\omega$ is the cyclic frequency of oscillations $$ \omega = \sqrt{\frac{g}{l}} $$ Let's take the derivative of $x(t)$: $$ v = \dot{x}(t) = \omega\cdot x_0\cos\omega t $$ Making the maximum speed equal to $$ \boxed{\upsilon_\text{max}=x_0\omega =x_0\sqrt{\frac{g}{l}}} $$
$$\upsilon_\text{max}=x_0\sqrt{\frac{g}{l}}$$