Soliton propagation direction can be engineered in optical fibers in the presence of high-order effects (HOEs). It is well known that Raman effects can decelerate the soliton. Here we investigate the manipulation of the deceleration or acceleration of soliton emitted from Airy pulse whose spectrum is imposed an initial quadratic phase modulation (QPM) in optical fibers in the absence of HOEs. We show that, under the action of the anomalous second-order dispersion (SOD) and Kerr nonlinearity, Airy pulse with QPM is able to emit soliton with acceleration or deceleration depending on whether the QPM is negative or positive, and at a rate that is determined by the magnitude of QPM. The reason is that the acceleration behaviors of incident Airy pulse is altered depending on whether SOD and QPM have the same or opposite signs. Our study shows the possibility of controlling and manipulating the soliton propagation and interaction in optical fibers without HOEs, by purposely choosing appropriate QPM parameter of an Airy pulse.