\documentclass{article} \usepackage{amsmath} \begin{document} The velocity of the mass as a function of time \( t \) is described by the equation: \[ v(t) = v_{\text{max}} \cdot \sin(\omega t + \phi) \] where: \begin{itemize} \item \( v_{\text{max}} \) is the maximum velocity (amplitude of velocity), \item \( \omega \) is the angular frequency of the oscillations, \item \( \phi \) is the initial phase. \end{itemize} The angular frequency \( \omega \) is related to the period \( T \) by the formula: \[ \omega = \frac{2\pi}{T} \] For a period \( T = 0.4 \) seconds: \[ \omega = \frac{2\pi}{0.4} = 5\pi \, \text{rad/s} \] If the mass starts moving from the equilibrium position with an upward velocity, the initial phase \( \phi \) is 0. Then, the velocity equation becomes: \[ v(t) = 0.8 \cdot \sin(5\pi t) \] \end{document}