Jem no longer cares if Atticus has special talents, because he does have one—he is the deadest shot in Maycomb County.
Jem and Scout think that their father is boring because he is older than most other dads and he works a lot. When he doesn’t work, he seems more interested in reading than athleticism. He does play with them, but not really the rough stuff. He is not a young man.
Atticus was never too tired to play keep-away, but when Jem wanted to tackle him Atticus would say, "I'm too old for that, son." (Ch. 10)
It sometimes seems as if Scout is closer to her father than Jem. After all, she is bookish, liking to read more than football. She is a tomboy, but also a bookworm. Jem, at this age, is pulling away.
At this point in the book, the children get a wakeup call. They learn that their father is not only capable, but important. When he shoots the mad dog, sick with rabies and dangerous to the whole town, they learn that their father is doing something no one else can do. It is then that Jem makes a realization.
"Naw, Scout, it's something you wouldn't understand. Atticus is real old, but I wouldn't care if he couldn't do anything- I wouldn't care if he couldn't do a blessed thing." (Ch. 10)
Jem is proud of his father for his physical achievement. He has proven himself to his children, but the people of the town knew he could do this. After all, Heck Tate passed the shot to Atticus, who hadn’t held a gun in years, knowing that only Atticus could make the shot.
The metaphor here is not lost on the reader. Atticus is the only thing standing between Maycomb and the danger. He is the only one brave enough, and the only one capable enough, of defeating it. The law passes him the torch. He is designated, by the legal authority, as the one to protect Maycomb. The mad dog is a symbol. It stands, metaphorically, for racism. Atticus is the only one who can defend the town from it. He will do it again, in the trial of Tom Robinson. The weapon has been passed.
Here is the scene from the film adaptation where Jem realizes his father's talent: