Let$$s assume that gravity is $(0,-10)$ and is the only applied force on the particle.

$($a$)$ Define C$($x$)$ and $$C$($x$).$ Multiple solutions will be acceptable.

$($b$)$ Let the current position of the red partial be $[1\mathrm{.}0,0\mathrm{.}0]$T $.$ Is it legal?

$($c$)$ Let the current velocity of the red partial be $[0\mathrm{.}1,2\mathrm{.}0]$T $.$ Is it legal?

Let$$s assume the feedback coefficients used to correct the particle are ks $=1\mathrm{.}0,$ kd $=0\mathrm{.}2.$

$($d$)$ Compute $$C$/$x and $$C$/$x$.$

$($e$)$ Compute the Lagragian multiplier $.$

$($f$)$ Compute the constrained force f $.$

$2$