A skier starts from rest at the top of a 20 degree incline and skis in a straight line to the bottom of the slope, a distance d (measured along the slope) of 400 m. If the coefficient of kinetic friction between the skis and the snow is 0.2, calculate the skier's speed at the bottom of the run.

Explanation:

Loss of potential energy = mgh.

h = d sin 20

= 400 sin20 = 136.8 m

Loss of potential energy = m x 9.8 x  136.8

= 1340.64 m

negative work done by friction = μ mg cosθ x d

= .2 x m x 9.8 x cos 20 x 400

= 736.72 m

Net loss of potential energy = 1340.64 m - 736.72 m

= 603.92 m

= gain of kinetic energy = 1/2 m v²

1/2 m v² = 603.92 m

v² = 1207.84

v = 34.75 m /s .

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.