A wave on a string is described by `y(x,t)=A cos (kx-omegat)`. (a) Graph `y,v_(y)`, and `a_(y)` as function of x for time t=0. (b) consider the following points on the string: (i) x=0,(ii) `x=pi//4k`, (iii) `x=pi//2k`, (iv)`x=3pi//4k`, (v)`x=pi//k`, (vi)`x=5pi//4k`, (vii)` x=3pi//2k`, (viii) `x=7pi//4k`. For a particle at each of these points at t=0, describes in words whether the particle is moving and in what direction, and whether the particle is speeding up, slowing down, or instantaneously not acceleraring.

Explanation

`y =A sin(omegat -kx),` Particle velocity
`v_p (X,t) =(delta y)/(delta t) = omegaA cos (ometat -kx)`
and particle acceleration
`a_p (x,t) = (delta^2y)/(deltat^2) =- omega^2 A sin(omegat -kx)`
i) `t =0 ,x=0 : v_p =+omegaA and a_p =0` i.e., particle is moving upwards but its acceleration is zero.
ii) At `t =0, x = pi/(4k), V_p =(A omega)/(sqrt2)`, upward and
`a_p =(omega^2A)/sqrt2` upward