When Homer placed the rocket on his fence and lit it his friend asked "Should we get behind somethin'?" That's when the rocket explodes. When you see them fly back you don't need to slow motion it because you can clearly see the wires attached to their backs.
Homer Hickam's father's name is actually Homer Hickam Sr.; in the movie, the father's name was changed to John. The assumption would be that the name was changed in the movie to avoid confusion between father and son.