It is given in the question, initial momentum of the particle-A is `p`.
Also, the momentum of particle -B will be zero due to its initial position at rest.
Now, after the collision, assuming p1 and p2 be the linear momentum of the particles A and B.
Now, Impulse imparted, I = change in momentum, `delta p`.
Impulse for particle-B  = I         .....(i)
Impulse for particle-A = p - p2    or  p2 = p - I  ..........(ii)     
  (Final momentum - initial momentum for particle -A).
Coefficient of restitution,

            Difference of final velocities after collision      v2 - v1      mv1 - mv2       p1 - p2
   e =   --------------------------------------------------------   =  ---------  =  -------------  =   ---------- .....(iii)
            Difference of initial velocities                                u                 mu                p        
 
  (where, m is the mass of the each particle and `u` is the velocity of particle-A before collision and v1 and v2 be the velocities after collision of particle A` and `B`).
Therefore, 
From eqns. (i), (ii) and (iii), we get to know,

             p1  - p + I         I -  p + I        2 I     
    e =   .................  =   ...............  =   .....  - 1
                  p                      p               p