I think there is something wrong in the formulation of this question.
With Impulse defined as
#vec J = int_(t = a)^b vec F(t) \ dt #
#= int_(t = a)^b vec dot p (t) \ dt = vec p(b) - vec p(a)#
then the Impulse on the object at t= 1 is
#vec J = int_(t = 1)^1 vec F(t) \ dt = vec p(1) - vec p(1) = 0#
It may be that you want the total impulse applied for #t in [0,1]# which is
#vec J = int_(t = 0)^1 vec F(t) \ dt = vec p(1) - vec p(0) qquad star#
To evaluate #star# that we note that if the rate of change of kinetic energy #T# is constant, ie:
#(dT)/(dt) = const #
then
#T= alpha t + beta#
#T(0) = 8 implies beta = 8#
#T(4) = 136 = alpha(4) + 8 implies alpha = 32#
#T= 32 t + 8#
Now #T = abs(vec p)^2/(2m)#.
#implies (vec p * vec p)= 4(32 t + 8)#
#vec p = 2sqrt( (32 t + 8)) hat p#
and
#vec p(1) - vec p (0)#
# = (2sqrt( (32 + 8)) - 2sqrt( 8) )hat p#
# = 4(sqrt( 10) - sqrt( 2) )hat p# N s
