A two state ratchet model is used to study the cooperative bidirectional motion of molecular motors on cytoskeletal tracks with randomly alternating polarities. Our model is based on a previous proposed model [Badual et al., Proc. Natl. Acad. Sci. U.S.A. 99, 6696 (2002)] for collective motor dynamics and, in addition, takes into account the cooperativity effect arising from the elastic tension that develops in the cytoskeletal track due to the joint action of the walking motors. We show that this additional cooperativity effect leads to a dramatic reduction in the characteristic reversal time of the bidirectional motion, especially in systems with a large number of motors. Our model succeeds in explaining recent experimental results which have been in contrast to previous theoretical predictions.