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The main factor that is different between the moon and Earth is the force of gravity. The moon's gravitational force is only 1/6th of that on Earth. This would present an issue in the gravitational acceleration of the football when thrown. The ball would accelerate at 1/6th the speed meaning it would have much more air time which would make the game different.
Another impacting factor would be the lack of atmosphere on the moon. This lack of air would present an absence of air resistance which would make the ball travel somewhat faster (without regards to difference in speed due to less gravity) from one point to another.
According to Newton's second law, force is equal to mass times acceleration. If it worked out perfectly on space and we were able to increase the mass of the ball by the same amount the acceleration decreased due to less gravity, ideally the same amount of force would have to be exerted on the ball to do the same thing you would do on Earth.
Due to less friction, adding extra mass to the players' equipment to make them weigh more might make running easier by increasing friction. But that could also make it more difficult to start moving in the first place so it would probably need to be tested to see what works best.
According to Newton's third law, for every action there is an equal and opposite reaction. This means that on the mean,as on Earth, when a player threw a ball the ball would exert the same force on the player. Or when the player pushes off the moon's surface, the moon pushes back with the same amount of force because Newton's laws are always applicable.
Brief History and Description of Football
While football traces its origins to rugby and soccer, the history of American football has resulted in a game that substantially differs from them.
Also, due to less friction because of the change in gravity, it would be significantly difficult for players to get a good grip on the surface on the moon which would make running hard. This is because they aren't being pulled down with as much force as usual so they wouldn't have as good of leverage when pushing off one foot to take a step.
Transformation to Moon Football
The game is played on a 100-yard field with two end zones, each 10 yards deep, between two teams consisting of 11 players on offense and 11 players on defense.Penalties play a large part in each football game. The most common penalties are holding, pass interference, false start, face mask, and unsportsmanlike conduct.
Although many things would change on the moon, three things stay reasonably constant: Newton's laws. According to the first law, things would still keep moving until a force acted on them. So, in this case, the ball would keep moving until either gravity or another player stopped/slowed it. Still gravity would take longer to stop the object than if it were here on Earth.
Several changes would have to be made to football to successfully play it on the moon. Due to 1/6th the Earth's gravity you would need to make the ball six times as massive as typical. Mass of the ball wouldn't change in anyway by using the ball on the moon but due to the amount of gravity pulling on it would make it weigh six times less so increasing the mass by six times would make it equal to the weight of a football on our planet.