In The Great Gatsby, how do the motifs of violence, colours, race, and sports relate to Tom Buchanan and Jay Gatsby?
Tom has violent tendencies. When Nick accompanies Tom and Myrtle to an afternoon party in the city, Tom breaks Myrtle's nose as she keeps repeating Daisy's name. If there is a color associated with Tom it is green, the color of money. Green is symbolic for different reasons in the novel, but with Tom, the symbolism has to do with money. Tom has racist theories on civilization and expresses them proudly. When Nick first visits Tom and Daisy (and Jordan) in Chapter One, Tom describes some racist literature he's been reading:
“Well, these books are all scientific,” insisted Tom, glancing at her impatiently. “This fellow has worked out the whole thing. It’s up to us, who are the dominant race, to watch out or these other races will have control of things.”
Tom still shows signs of his powerful athleticism having once been "one of the most powerful ends that ever played football at New Haven."
Gatsby is not prone to, or comfortable with, violence. However, his naive and idealistic actions do involve violence. Remember that Gatsby is involved with Meyer Wolfsheim, doing business in bootlegging and possibly illegal drug trade. Violence can, at least occasionally, accompany this kind of illegal business. But Gatsby is on the business side of things and distances himself from the dirty part of the business. Gatsby's action do also play a part, indirectly, in Myrtle's death. So, he has associations with violence but he does not embody violence the way that Tom does.
The color green is also significant for Gatsby for similar and different reasons. Gatsby is interested in money, as Tom is. Acquiring wealth is a way to better himself and to put himself in a position on par with Daisy's social status and Gatsby's idealistic vision of her. Later in the novel, Gatsby remarks that Daisy's voice is "full of money." Of this, Nick remarks:
That was it. I’d never understood before. It was full of money—that was the inexhaustible charm that rose and fell in it, the jingle of it, the cymbals’ song of it. . . . high in a white palace the king’s daughter, the golden girl. . . (Chapter 7)
But green also has a more general symbolic meaning for Gatsby. Gatsby stares at the green light at the end of the dock near Daisy's house and, for Gatsby, that light becomes a symbol of Daisy herself. In fact, Gatsby conflated his ideal vision of Daisy with the idea of money, two things which signify wealth, achievement, and a prize with some deeper meaning such as a Holy Grail.
In the last chapter, Gatsby's father shows Nick Gatsby's regimen he had when he was a boy. The regimen included dumbbell exercises, wall-scaling, studying, baseball and sports, etc. Given that Gatsby, as an adult, continued to improve himself to the point where/when he would be worthy of Daisy's affection, it is safe to assume he continued the same or a similar kind of self-improvement, including mental and physical exercises.