Some things are still a bit unclear. Does the moving object come to a stop? Is it a head-on 1-dimensional problem?
If it is a head-on 1-dimensional problem (object comes to a stop or goes back in opposite direction), then the deflection of the stationary rod is simply the resultant deflection from absorbing the kinetic energy. This is analogous to the amount of energy a compression spring will absorb over a deflection, E=0.5*k*x^2. There is an equivalent equation for a fixed cantilever beam. Looks like one of my esteemed colleagues already gave it to you.
You must decide what is "negligible" deflection. Give it a quantity. Is it a fixed number, or is it just sufficient that the rod does not fail?