At the moment `t=0` a point starts oscillating along the `x` axis according to the lasw` x=a sin omegat t`. Find:
`(a)` the mean value of its velocity vector projection `( : v_(x) : )`,
`(b)` the modulus of the mean velocity vector `|( : v : )|`,
(c) the mean value of the velocity modulus `( : v : )` averaged over `3//8` of the period after the start.

For a particle oscillating along x-axis according to equation x =A sin omegat The mean value of its velocity averaged over first 3//8 of the period is

A point moves along th e x axis according to the law x=a sin^(2)(omegat-pi//4) Find. (a) the amplitude and period oscillations, draw the plot x(t), (b) the velocity projection upsilon_(x) as a function of the coordination x , draw the plot upsilon_(x)(x) .

A point oscillates along the x axis according to the law x=a cos (omegat-pi//4) . Draw the approximate plots (a) of displacemetn x , velocity projection upsilon_(x) , and acceleration projection w_(x) as functions of time t , (b) velocity projection upsilon_(x) and acceleration projection w_(x) as functions of the coordiniate x .

Using the Maxwell distribution function, calculate the mean velocity projection (v_x) and the mean value of the modules of this projection lt lt |v_x| gt gt if the mass of each molecule is equal to m and the gas temperature is T .

At the moment t=0 a particle starts moving along the x axis so that its velocity projection varies as v_(x)=35 cos pi t cm//s , where t is expressed in seconds. Find the distance that this particle covers during t=2.80s after the start.

A radius vector of a point A relative to the origin varies with time t as r=ati-bt^2j , where a and b are positive constants, and I and j are the unit vectors of the x and y axes. Find: (a) the equation of the point's trajectory y(x) , plot this function, (b) the time dependence of the velocity v and acceleration w vectors, as well as of the moduli of these quantities, (c) the time dependence of the angle alpha between the vectors w and v, (d) the mean velocity vector averaged over the first t seconds of motion, and the modulus of this vector.

A small body is thrown at an angle to the horizontal with the initial velocity v_0 . Neglecting the air drag, find: (a) the displacement of the body as a function of time r(t) , (b) the mean velocity vector ltlt v gtgt averaged over the first t seconds and over the total time of motion.