$(45\mathrm{.}1)$ Two particles, $M_1=5kg$ and $M_2=15kg,$ move in one dimension and experience a completely elastic collision. Just prior to the instant of collision, block $1$ has velocity $v_1=3\frac{m}{s}$ while block $2$ has $v_2=1\frac{m}{s}.[$Positive velocity means "forward, to the right."$]$

$($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$)$ Compute the velocity of each block the instant after the collision.

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

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

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

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

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

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

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

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

$($g$)$ Compute the velocify of the centre of mass of the system immediately after the collision.

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

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

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

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

$(45\mathrm{.}2)$ Without doing any computation, determine the final velocities of the blocks if initially $v_1=0\frac{m}{s}$ and $v_2=2\frac{m}{s}.$