$(47\mathrm{.}1)$ Two particles, $M_1=5kg$ and $M_2=15kg,$ move in one dimension and experience a completely inelastic collision. Just prior to the instant of collision, block $1$ has velocity $v_1=3\frac{m}{s}$ while block $2$ has $v_2=1\frac{m}{s}.$

$($a$)$ Compute the initial momentum of

$($i$)$ block $1,$

$($ii$)$ block $2,$ and

$($iii$)$ the twoblock system.

$($b$)$ Compute the initial kinetic energy of

$($i$)$ block $1,$

$($ii$)$ block $2,$ and

$($iii$)$ the two$-$block system.

$($c$)$ Compute the velocity of the centre of mass of the system prior to the collision.

$($d$)$ Determine the velocity of the block $1-2$ aggregate an instant after the collision.

$($e$)$ Compute the final momentum of the aggregate.

$($f$)$ Compute the final kinetic energy of the aggregate.

$($g$)$ Ascertain how much kinetic energy was lost in the collision.

$($h$)$ Compute the impulse received by

$($i$)$ block $1,$

$($ii$)$ block $2,$ and

$($iii$)$ the system. Comment.

$(47\mathrm{.}2)$ Suppose instead that the particles are initially moving in opposite directions: $v_1=3\frac{m}{s}$ and $v_2=-1\frac{m}{s}.$ Determine the final velocity of the aggregate and the amount of kinetic energy lost in this collision.