Q$2.$ Figure $2$ shows $5-kg$ block, B resting on $8-kg$ block, A which is placed on a frictionless surface where the inclination angle, $=20.$

i$)$ Sketch Free Body Diagrams of each block. Write dynamic equilibrium equations for each block at each direction. Find the accelerations $a_{\frac{A}{G}},a_{\frac{B}{G}},$ and $a_{\frac{B}{A}}$ by sketching the Kinetic Diagram. $($Vector Mechanics Approach through Newton's $2^{nd}$ Law$)$

ii$)$ Find the velocities $v_{\frac{A}{G}},v_{\frac{B}{G}},$ and $v_{\frac{B}{A}}$ when block, $B$ starts motion from rest $($initial position, $1)$ and slides on the inclined surface of block, $A$ by $2$ meters $($final position, $2).$

iii$)$ How many meters in how much time does block, A move with respect to the ground when block, $B$ moves $2$ meters on the inclined surface of $A?$

iv$)$ Write the Work Equation for the system, $W_{A2}$ and find the velocities $v_{\text{A}\text{V}\text{G}}$ and $V_{\frac{B}{G}}$ by using Energy Approach when block, B starts motion from rest $($initial position, $1)$ and slides on the inclined surface of block, $A$ by $2$ meters $($final position, $2).$ Show that the velocities are consistent with the ones you have found in item $ii).$

v$)$ Show that:

${?}_{\text{r}\text{a}\text{r}\text{r}}^{1W{2}^{\text{s}\text{y}\text{s}\text{t}\text{e}\text{m}}}=-$Delta $U$

$1W{2}^{\text{s}\text{y}\text{s}\text{t}\text{e}\text{m}}=T_2^{\text{s}\text{y}\text{s}\text{t}\text{e}\text{m}}-T_1^{\text{s}\text{y}\text{s}\text{t}\text{e}\text{m}}$

vi$)$ Use the definition of Force; i$.$e$.,$ Fexternal $*t=L^{\text{s}\text{y}\text{t}\text{e}\text{m}}$ to find the time interval when block, $B$ starts motion from rest $($initial position, $1)$ and slides on the inclined surface of block, $A$ by $2$ meters $($final position, $2).$ Show that it is consistent with what you have found in item iii$).$

vii$)$ Show that the equation,

$F^{\text{e}\text{x}\text{t}\text{e}\text{r}\text{n}\text{a}\text{l}of\text{b}\text{l}\text{o}\text{c}\text{k}of\text{i}\text{n}\text{t}\text{e}\text{r}\text{e}\text{s}\text{t}}*t=L^{\text{b}\text{l}\text{o}\text{c}\text{k}of\text{i}\text{n}\text{t}\text{e}\text{r}\text{e}\text{s}\text{t}}$

can also be used for each block separately in order to find the unique interval of time when block, $B$ starts motion from rest $($initial position, $1)$ and slides on the inclined surface of block, $A$ by $2$ meters $($final position, $2).$