Existence of dense velocity curve on connected manifold

Question

Let $M$ be a non-empty, connected manifold. Show that there exists a differentiable curve $\gamma : \mathbb{R} \to M $ , so that the image of the velocity curve $\dot{\gamma} :\mathbb{R} \to TM$ is dense on $TM$.

This is a problem on chapter 11 "Second order differential equations and spays" of Bröcker's book.

I have no clue on how to solve this, so any hint will be very appreciated. Thank you so much!