Disregarding air friction, what force(s) act on a bullet shot from a rifle as it moves through the air?

The gravity.
In fact, the gravity will "pull" the bullet towards the ground, and the magnitude of the force is equal to the weight of the bullet:

where
m is the mass of the bullet
is the acceleration of gravity

Due to the presence of this force, the bullet will have a parabolic motion, which consists of two independent motions on the horizontal axis and on the vertical axis:
- on the horizontal axis, the bullet moves by uniform motion with constant speed
- on the vertical axis, the bullet moves by uniformly accelerated motion, with constant acceleration g towards the ground.

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