A mass $m$ is attached to a spring with spring constant $k$ on a frictionless table. We stretch and release it a distance $x_0$ away from the resting position. $($a$)$ Sketch the position, velocity, and acceleration plots $($time on $x-$axis$).($b$)$ What is the period if: $($i$)$ the mass is doubled, $($ii$)$ or the spring constant is quadrupled, and $($iii$)$ or we double $x_0$ but the mass and spring constant remain unchanged? $($c$)$ On one plot, sketch the total energy, kinetic energy, and elastic energy with time on the $x-$axis.

$($d$)$ When the mass is at the position $-\frac{x_0}{3},$ what is the velocity? $($e$)$ At what distance will the speed of the mass be half of its maximum speed?