A ball is thrown vertically upward from the ground with speed v0 and is subject to a quadratic drag with magnitude cv2 . a) Show that the maximum height the ball reaches is given by ymax = v 2 ter 2g ln v 2 ter + v 2 0 v 2 ter . b) Show that the speed of the ball when it returns to the ground is given by vterv0 p v 2 ter + v 2 0 . Discuss the case when there is a lot of air resistance, and the case when there is very little.

that would be because of both gravity and friction. so     c

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.