What is the optimal angle when throwing a stone on a slope? The question what-is-the-optimum-angle-of-projection-when-throwing-a-stone-off-a-cliff was asked and answered a while back.  This one has a much cleaner answer.  Now you are on a uniform slope, a line through the origin that is not horizontal and you want to throw a stone as far as possible.  As a function of the angle of the slope, what angle should you throw at?  Usual assumptions:  uniform gravity, no friction.

I am surprised no one seems to have answered this.
I get $$\pi/4 + \alpha/2$$.
If the line is given by $y = x \tan \alpha$, with $\alpha$ acute and we throw from the origin at an angle $\theta$ from the x-axis, at velocity $1$, then we have that the projectile satisfies, assuming gravity $g=2$ (in appropriate units)
$\displaystyle y = t\sin\theta - t^2$, $\displaystyle x = t\cos \theta$
The time at which it intersects the line again is given by
$\displaystyle t\sin\theta - t^2 = t \tan\alpha \cos \theta$ and so
$\displaystyle t = \sin \theta - \tan \alpha \cos \theta$
It is enough to maximize the horizontal distance travelled by the projectile, which is given by
$\displaystyle \cos \theta (\sin \theta - \tan \alpha \cos \theta)$
With little manipulation, we need to maximize
$\displaystyle  \sin(2\theta - \alpha)$
which gives the result.
