Can the mother of my child move out of state

Generally, the Court retains jurisdiction over the case and children in most dissolution or custody matters to enforce or modify orders, including parenting plans and time-sharing.  Additionally, parents are not allowed to move out of state with a child once the Court has jurisidction over the child unless there is a modification of the time-sharing and parenting plan and court approval.  With that said, if the mother relocates with the child without permission then she will be in violation of the court order and you can proceed with a motion for contempt and enforcement of the time-sharing.  If the Judge does permit the mother to relocate and the time-sharing is changed, based upon that relocation, then there will no longer be a 50/50 time-sharing, thus the child support will then change, and so if the child is with the mother the majority of the time then you will likely end up paying more in child support and see your child less.  I would suggest consulting with an attorney to review all of your options and the possible outcomes.