Cut a 40-by-76-inch rectangle of terrycloth. Cut two 12-by-40-inch and two 12-by-76-inch strips of terrycloth. Cut a 36-by-36-inch square of vinyl. Cut four 12-inch pieces of elastic.
Sew the short ends of the 76-inch terrycloth strips to the short ends of the 40-inch strips to form the sides of the mattress cover. Turn under one long edge of the connected strips 1/2 inch and use a zigzag stitch to sew along the edges.
Pin the large terrycloth rectangle (the top of the mattress cover) to the unfinished edges of the side strips. Sew the top to the sides.
Center a piece of elastic at the bottom corner of the side edge of the cover. Stretch the elastic around the corner and pin it in place. While keeping the elastic stretched, zigzag the elastic to the edge. Repeat for the remaining three corners.
Sew one side of a Velcro fastener to each corner of the vinyl square.
Lay the vinyl piece on the underside of the mattress cover, positioning it so that the vinyl square is centered across the width of the cover, and the bottom edge of the vinyl is 18 inches from the bottom edge of the terrycloth top.
Pin the other half of the Velcro fasteners to the terrycloth so that the Velcro on the vinyl matches up with the Velcro on the terrycloth. Sew the Velcro pieces to the terrycloth.