Assuming that the earth is a perfect sphere with radius a, we start our journey in the point $(R, \phi, \theta) = (a, \pi/2, 0)$ (Spherical coordinates) and we travel North-West with constant speed v.
I want to find the function $\phi(t)$ and $\theta(t)$ that describe the path. The question at first is, why for $\phi(t)$ the differential equation
$$a \frac{d\phi}{dt} = -\frac{v}{\sqrt2}$$ holds.
Understanding this should give information to set up a differential equation for $\theta(t)$.
The problem is that I don't understand the differential equation that holds for $\phi(t)$, I thought it might have something to do with the fact that $sin(\frac{\pi}{4}) = cos(\frac{\pi}{4})=\frac{1}{\sqrt2}$. Solving it, I find $\phi(t) = \frac{-v}{a\sqrt2}t+C$ and because we have the starting point $(a, \pi/2,0)$, we see that $C = \pi/2$.
How should I read the differential equation for $\phi(t)$ and how can I set up one for $\theta(t)$?