A $0\mathrm{.}50$ kg object rests on a frictionless horizontal countertop, where it is attached to a massless spring whose k$-$value equals $25\mathrm{.}0$ N$/$m$.$ Let x be the displacement, where

x $=0$

is the equilibrium position and

x $>0$

when the spring is stretched. The object is pushed, and the spring compressed, until

xi $=4\mathrm{.}00$ cm$.$

It then is released from rest and undergoes simple harmonic motion.

$($a$)$

What is the object's maximum speed $($in m$/$s$)$ after it is released?

m$/$s

$($b$)$

How fast is the object moving $($in m$/$s$)$ when the spring is momentarily compressed by $1\mathrm{.}80$ cm $($that is$,$ when

x $=1\mathrm{.}80$ cm$)?$

Mechanical energy is conserved in this system, and the gravitational term remains unchanged $($since all motion is horizontal$).$ Write an expression for mechanical energy that includes the kinetic energy and the potential energy of the spring. Which term$($s$)$ can be ignored when the spring is compressed and the object at rest? Use the remaining terms, and the given quantities, to solve for the maximum speed. m$/$s

$($c$)$

How fast is the object moving $($in m$/$s$)$ whenever the spring is extended by $1\mathrm{.}80$ cm $($that is$,$ when passing through

x $=+1\mathrm{.}80$ cm$)?$

m$/$s

$($d$)$

Find the magnitude of the displacement $($in cm$)$ at which the object moves with one$-$half of the maximum speed.

$|$x$|=$ cm