Aspring with a spring constant k of 320 newtons per meter is loaded with a 5-kg weight and allowed to reach equilibrium. it is then displaced 1 meter downward and released. if the weight experiences a damping force in newtons equal to 64 times the velocity at every point, find the equation of motion. y(t)= , where t is time and y(t) is displacement in time.

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.

maria has a savings account that is earning simple interest. if she started with $200 in the account and has earned a total of $8 in interest in two years, what is the rate of the account? use the formula  .