In this contribution we study the "Bando-follow the leader" car-following model, a second order ordinary differential equation, for its well-posedness. Under suitable conditions, we provide existence and uniqueness results, and also bounds on the higher derivatives, i.e., velocity and acceleration. We then extend the result to the "reaction" delay case where the delay is instantiated in reacting on the leading vehicle's position and velocity. We prove that the solution of the delayed model converges to the undelayed when the delay converges to zero and present some numerical examples underlying the idea that it is worth looking in more details into delay as it might explain problems in traffic flow like "phantom shocks" and "stop and go" waves.