Infinite string wave equation

Infinite string at rest for t<0, has instantaneous transverse blow at t=0 which gives initial velocity of $V \delta ( x - x_{0} )$ for a constant V. Derive the position of string for later time.

I thought that this would be $y_{tt} = c^{2} y_{xx}$ with $y_{t} (x, 0) = V \delta ( x - x_{0} )$, and $y(x,0) = 0$. So use the d'Alembert solution $y = f(x + ct) + g(x - ct)$. Then applying these gets the forms of f, g.

Is this correct? I am a bit nervous because I am self-teaching some of this wave equation stuff and I am not good at applying the theory to a practical question