An amusement apparatus has a platform, pulleys mounted below the platform, a cable extending around the pulleys, a motor coupled to a pulley to cause the cable to move, a passenger vehicle supported by wheels on the platform, and a swing coupling having a horizontal arm coupled to the passenger vehicle and a vertical element coupled to the arm and coupled to the cable. The cable traverses a path having both left and right turns. In one embodiment, the platform defines a continuous slot located above and substantially following the path. In another embodiment, the platform defines a continuous slot with first portions above but laterally parallel the cable on one side of the cable, second portions above and laterally parallel the cable on the other side of the cable, and third portions crossing over the cable and connecting the first and second portions. Different swing coupling arrangements are described.