What were three scientific problems that the boys had to solve?
The film "October Sky" tells the story of a group of American boys in the late 1950s who are inspired to build and launch their own rockets. Their main challenge is the fact that they live in a coal-mining town in West Virginia; this limits their technological and mechanical resources and sets them at odds with some of the town's residents. Much of the film's conflict arises from man-vs-machine and man-vs-nature problems, in that the boys are pursuing ways of buildling better rockets, as well as man-vs-man conflicts, when their rocket-building incurs the anger of their parents and other authority figures. The memoir upon which it was based, Rocket Boys, contains additional scientific details.
One problem the boys addressed was the question of what fuel to use in the rockets. The exact type of fuel used in a rocket makes a considerable contribution to the rocket's successful flight, altitude, stability and other properties. If the fuel is too reactive, and goes off all at once, the rocket will simply explode. The fuel also needs oxygen to burn; if the rocket goes high enough that the amount of oxygen in the atmosphere decreases, the rocket's thrust will decrease as well. This is why "professional" rockets carry their own oxygen. The boys experimented with a variety of fuels, including a homemade concoction called zincoshine; a combination of zinc, sulfur and moonshine.
Two more problems are introduced after one of their rockets falls astray and is accused of starting a forest fire. The scientific problem that they boys must solve comes in two parts.
First, they lacked the knowledge of how to calculate a rocket's trajectory; Homer fortuitously finds this in a book. Using this math, he is able to make a hypothesis about where they will find the lost rocket. His prediction indicates that the rocket could not have caused the fire.
When they go to the predicted location, the rocket is not found. This initiates the second part of the problem; revising the math, and revising the hypothesis. Homer trusts that the math he used was fundamentally sound, so he looks to variables; he knows that the wind on the day of the launch was westward, and since the rocket fell for several seconds, the wind may have had enough time to push it away from the "mathematically correct" landing location. Rather than doing more math, he simpy looks west, in the direction of the wind on the launch day, and sees the rocket, confirming his modified hypothesis.