In the movie October Sky, what motivated the boys' learning?

The boys in October Sky each have their own motives for learning rocket science. Homer Hickam, the story's central character, sees science and rockets as a way to avoid a life working in the coal mines. Homer's father is a mine foreman who sees a future as a miner for his son. Homer has other ideas. He receives his initial inspiration when the Russians launch Sputnik, the first artificial satellite. After viewing the satellite in orbit in the sky above his town, Homer realizes that he has an opportunity to make a life for himself in rocket science. He decides to build a rocket for the science fair. He is further inspired by his math teacher, Miss Riley, who encourages him to pursue his passion for math and science. Miss Riley serves as a constant inspiration for Homer and the other boys. She is a dedicated teacher who does all she can to help her students avoid a life in the coal mines. Homer also wants to avoid the mines and escape his small town. This repeatedly causes conflict with his father. Homer summarizes his motives clearly in an argument with his father:

If I go on to win at Indianapolis, I can go to college, maybe even get a job at Cape Canaveral. There's nothing here for me. The town is dying! The mine is dying! Everybody here knows that but you!

The motives of the other three boys who work on the rockets are not addressed quite as much. Quentin Wilson, the school's math nerd, also has an interest in rockets. The source of his interest is not readily addressed in the movie. It is clear that Quintin does not see himself destined for a life of a coal miner. He is too much of an intellectual. It is likely that he is just happy to have someone interested in math and rockets who he can work with.
Homer's two other friends, Roy Lee Cooke and Sherman O'Dell, are more unlikely participants in the rocket project. They do not have a particular predilection for science, but they are happy to help their friend. In many ways, the rockets give them something to do. It is an interesting break from the monotony of growing up in a small town with few diversions. In many ways, they look up to Homer. They see him as someone who can escape the fate of growing up with no future outside of coal mining. By helping Homer succeed, they give themselves hope that they can make their own futures too.