We can approximate the relationship between position $p(k),$ velocity $v(k),$ and acceleration $u(k)$ using the state$-$space model

ubrace$(\left[\begin{array}{c}p(k+1)\\ v(k+1)\end{array}\right]$ubrace${)}_{x(k+1)}=\left[\begin{array}{cc}1& T_s\\ 0& 1\end{array}\right]$ubrace$(\left[\begin{array}{c}p(k)\\ v(k)\end{array}\right]$ubrace${)}_{x(k)}+\left[\begin{array}{c}0\\ T_s\end{array}\right]u(k)$

where $T_s$ is the sample$-$time.

a$.$ Write the equations for $p(k+1)$ and $v(k+1)$ in terms of $p(k),v(k)$ and $u(k).$ What happens to these relationships as $T_{s}0?$

b$.$ For the initial condition $x(0)=(0,5)$ and input $u(k)=0,$ compute $x(1)$ and $x(2)$

by hand. What does this mean about the position and velocity at time instances $k=1$ and $k=2.$

c$.$ For the initial condition $x(0)=(0,5)$ and input $u(k)=1,$ compute $x(1)$ and $x(2)$ by hand. What does this mean about the position and velocity at time instances $k=1$ and $k=2.$