Optimizing Projectile Distance on an Inclined Plane

[leitz]

Homework Statement

"A person stands at the base of a hill that is a straight incline making an angle ϕ with the horizontal. For a given initial speed v₀, at what angle θ (to the horizontal) should objects be thrown so that the distance d they land up the hill is as large as possible?"

I've looked up the same question on the internet, but the answers are either the formula (1/2arctan(-cotϕ) = θ) or π/4+ ϕ/2. I already figured out how to get 1/2arctan(-cotϕ) = θ, but I do not know how they got they π/4+ ϕ/2. Also, which is the correct answer?

Homework Equations

1/2arctan(-cotϕ) = θ

 

[Delphi51]

The π/4 + φ/2 formula predicts 45 degrees when φ=0, which is certainly correct. The arctan formula doesn't work at φ=0. At φ=45 degrees, the first formula predicts a reasonable 67.5 degrees, while the arctan formula seems to say -22.5 degrees. Better check my calc; I make mistakes.

The second formula is pretty well known. No doubt you could find a derivation of it on the web. Try searching for "Projectile motion on an incline". Of course it is always better to figure it out yourself if you have the time.