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The effect of acceleration - variable forward velocity, have been investigated on the steady-state and transient responses of turning vehicles and the resulted path of motion of the vehicle. Comparing the two mentioned responses of the vehicle could be used to prove that there is a negligible difference between the steady-state and transient centre of rotation of the vehicle in engineering applications. Dynamics of a vehicle with a constant steer angle and variable forward velocity have been analysed and presented as the proof. It has been shown that it is possible to predict the dynamics of vehicles using their steady state responses within acceptable engineering approximations. More specifically, we determine the dynamic rotation centre of vehicles and compare them with steady-state values. The result would be essential to design autodriver algorithm for autonomous vehicles.