An archer shoots an arrow in the air, horizontally. However, after moving some distance, the arrow falls to the ground. Name and define the initial force that sets
the arrow in motion. Explain why the arrow ultimately falls down.

The name of the force is elastic (spring) force

Explanation:

The elastic force which is the restoration force of an elastic (spring like) object that tries to return to its initial non-stretched state built up by the pulling of the bow by the archer (using the muscles) pushes on the arrow which the archer releases the same time he leaves the bow string, and the arrow flies in the direction already pointed by the archer

The arrow ultimately falls down due to gravity forces that acts on all objects within the Earths gravitational field. However due to the speed of constant horizontal speed of the arrow and increasing downward speed of the due to the gravity force, the falling of the arrow looks disappointing.

what was the answer?

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.