Problem 1.22

The velocity of a particle moving in the positive direction of the \(x\) axis varies as \(v=\alpha \sqrt{x},\) where \(\alpha\) is a positive constant. Assuming that at the moment \(t=0\) the particle was located at the point \(x=0,\) find: (a) the time dependence of the velocity and the acceleration of the particle; (b) the mean velocity of the particle averaged over the time that the particle takes to cover the first \(s\) metres of the path.