When children get out of kindergarten they still need to learn coordination but they now get to the age where their hands and feet obey their mind. Now they need to build strength, speed, and do more detailed things to develop better coordination. That is where sports come in. The different sports give children a reason to run, throw things, and kick things without getting in trouble. As they grow up children learn to aim with their feet and hands to earn praise from their parents. As they grow older they learn to be the fastest and most precise in order to be the best and earn praise from everyone as well as have a sense of pride in their actions.
As ship and culture creator, you need to decide what sports your ship and culture emphasis. Here is a list of sports recognized by most people. True, many of these sports can’t be done very easily in the confines of a space ship, unless you count virtual reality rooms, but there are a lot of sports for you to chose from or adjust your own sport from a few already around. After all, kids are always trying new things and trying to be cool by thinking up a new sport. Bear in mind that one court or field can hold a lot of sports. Things like soccer and American football have the same lines on the field and often volleyball and basketball are played on the same court, just at different times. Many sports stadiums are designed to hold a concert one day, a football game the next, and a basketball tournament the next day. You could always have one large room with adjustable stadium seating and storage rooms with different mats. That way you could allow the children on your ship to learn a large number of sports and not get focused on one sport out of boredom.