Atire of radius r rolls in a straight line without slipping. its center moves with constant speed v. a small pebble lodged in the tread of the tire touches the road at t = 0. find the pebble’s position, velocity, and acceleration as functions of time.

i've seen solutions before, starting with breaking the motion into x and y coordinates, like so

v(x)=v-rw*cos(wt)

v(y)=v(1-cos(wt))

i do not conceptually understand how these equations are derived, use pictures if nessecary.

it is called kinetic energy

beta is electron from nucleus

something from nucleus has changed, but charge must be same for cons of charge

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.