Two particles A and $B$ are moving towards each other along a straight line on a smooth horizontal surface.

The particles collide.

Particle A has mass $4kg$ and particle $B$ has mass $6kg$

Before the collision particle A has speed $5m{s}^{-1}$ and particle $B$ has speed $3m{s}^{-1}$

After the collision particle $B$ has speed $1\mathrm{.}48m{s}^{-1}$

The coefficient of restitution between the two particles is $e$

$($a$)$ Find the value of $e$

Fully justify your answer.

$($b$)$ Find the magnitude of the impulse exerted on $A$ by $B$ during the collision.

$($c$)$ Assume that the particles exert a constant force of $672$ newtons on each other during the collision.

Find the time that the particles are in contact with each other.