Geodesics on $S^2$ Correct description of this problem is here: Geodesics on $S^2$ with specific Riemannian metric
I was asked to show the following:

Given a Riemannian metric on $S^2$, let $x \in S^2$ be the north pole, and let $V(\theta) \in T_xS^2$ be unit tangent vectors at $x$. Then there exists a constant $t > 0$ such that $\exp_x(tV(\theta))$ is the south pole for all $\theta$. 

The conclusion in this statement is really not intuitive for me, and I am not sure how to approach this problem. Could someone give a hint on this?

Hints: What are the geodesics of a round unit sphere?
Start at a point $x$ of a round unit $2$-sphere and walk at unit speed. How long does it take to reach the antipode $-x$?

By contrast, consider an arbitrary metric on a $2$-sphere. As a physical analogy, imagine the surface of a squash or a gourd, where the center of the stem end is the north pole and the south pole is the center of the blossom end. It's not obvious that a geodesic starting at the north pole ever reaches the south pole. Certainly, different geodesics between the poles have different lengths.
