The doubly special relativity (DSR) theories are suggested in order to incorporate an observer-independent length scale in special theory of relativity. The Magueijo-Smolin proposal of DSR is realizable through a particular form of the noncommutative (NC) spacetime (known as $\kappa$-Minkowski spacetime) in which the Lorentz symmetry is preserved. In this framework, the NC parameter $\kappa$ provides the origin of natural cutoff energy scale. Using a nonlinear deformed relativistic dispersion relation along with the Lorentz transformations, we investigate some phenomenological facets of two-body collision problem (without creation of new particles) in a $\kappa$-Minkowski spacetime. By treating an elastic scattering problem, we study effects of the Planck scale energy cutoff on some relativistic kinematical properties of this scattering problem. The results are challenging in the sense that as soon as one turns on the $\kappa$-spacetime extension, the nature of the two-body collision alters from elastic to inelastic one. It is shown also that a significant kinematical variable involving in heavy ion collisions, the rapidity, is not essentially an additive quantity under a sequence of the nonlinear representation of the Lorentz transformations.
