In a direct impact, loss in kinetic energy is given by
ΔK=M1M22(M1+M2).(V1−V2)2(1−k2)
with usual notations (except k). The quantity k will have dimensional formula:
[M0L0T0]
Dimension of k2 will be same as that of 1, i.e., [M]0[L]0[T]0.
Hence, dimension of k is [M]0[L]0[T]0.