I cannot understand this particular moment in the movie. What does it represent for the story and how is it connected to the storyline? It appears to me that it's outside of context. Maybe I missed something or, due to my inability to understand English, this particular Atticus' action looks random. According to shmook the mad dog is symbol of injustice. How? Why?
The mad dog is a metaphor for the madness within society. The instilled racism which forms much of the common narrative is released at this point.
Atticus is typically seen as a sensible man the way he tells his children to shoot tins rather than birds etc.
Upon killing the mad dog the "injustice" that you are referring to reflects the same treatments of black people.
The injustice is how quickly the dog was executed by Atticus and thus more widely how quickly Maycombe as a whole were to blame Tom Robinson for the rape of a white woman just because he is black.
So the dog is both a comment of wider society but also the inner turmoil of Atticus that leads him to side with Tom.
As a side note its often been cinematical symbolic of black people through the use of Animal symbols. King Kong represented Black people coming to America to steal the white woman.
I don't know if I agree with the symbolism answer above. Atticus has told Scout that fighting (physical violence) is wrong. He is also a lawyer who only uses persuasion in an attempt to free his client. However, the rabid dog is a direct threat to his family and society. He kills the dog to protect everyone. Scout then realizes that her father is not a coward.