Chef and Puppy

Chef is working as a puppy guide . Puppy walks along the $X$ axis. Puppy can choose either of two directions and any speed not exceeding $V$. Puppy can also stand without moving anywhere. The chef informs them that at time $t1$ in the point with co-ordintate $x1$ a bone will occur , at time $t2$ in the point with co-ordintae $x2$ - another one , and so on up to ($xn$, $tn$) . It takes almost