You are in a hot air balloon, 650 m above the flat texas plains. the earth's radius is about 6400 km. you look out toward the horizon. how far out can you see−that is, how far is your horizon?

91 km

Explanation:

Draw a picture.  The line of sight to the horizon forms a tangent line with the curvature of the Earth.  The radius line to that point forms a right angle with the tangent line.  So we can use Pythagorean theorem to find the distance.

c² = a² + b²

(6400 km + 0.650 km)² = (6400 km)² + d²

d ≈ 91 km

the closer the planet is to the sun, the larger the gravity

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.