Consider the well-known problem of a cart of mass m moving along the x axis attached to a spring (force constant k), whose other end is held xed. if we ignore the mass of the spring, then we know that the cart executes simple harmonic motion with angular frequency ! = p k=m. 1. assuming, that the spring has mass m, that it is uniform and stretches uniformly, show that its kinetic energy is 1 6mx_ 2. (as usual x is extension of the spring from its equilibrium length.) 2. write down the lagrangian for the system of cart plus spring. write down the lagrange equation of motion. find the frequency of the small oscillations.

dang that balls went far

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.

answera compound;