Runner’s High (Speed)

I find the following mind-boggling.

Suppose that runner $ R_0$ runs distance $ [0,d_0]$ with average speed $ v_0$ . Runner $ R$ runs $ [0,d]$ with $ d>d_0$ and with average speed $ v > v_0$ . I would have thought that by some application of the intermediate value theorem we can find a subinterval $ I\subseteq [0,d]$ having length $ d_0$ such that $ R$ had average speed at least $ v_0$ on $ I$ . This is not necessarily so!

Question. What is the smallest value of $ C\in\mathbb{R}$ with $ C>1$ and the following property?

Whenever $ d>d_0$ , and $ R$ runs $ [0,d]$ with average speed $ Cv_0$ , then we can guarantee that there is a subinterval $ I\subseteq [0,d]$ having length $ d_0$ such that $ R$ had average speed at least $ v_0$ on $ I$ .