How Does Projectile Angle Affect Range When Fired Up a Sloped Hill?

[Serenity16]

Homework Statement

A projectile is fired with initial speed v0 at an elevation angle (alpha) up a hill of slope (beta) (alpha > beta).

a) How far up the hill will the projectile land?
b) At what angle (alpha) will the range be a maximum?
c) What is the maximum range?

Homework Equations

x = v0tcos(alpha)
y = -(gt^2)/2 + v0sin(alpha)
r = (x^2 + y^2)^(1/2)

The Attempt at a Solution

I had thought that the solution to this problem might be as simple as finding where the line that represents the hill intersects with the line that represents the trajectory of the projectile. But I'm not sure how to get there from here. I would know how to solve this if the range I was concerned about was the range from where the projectile is launched to where it hits the ground again (i.e. y2 = 0), but this has me stumped. Anything to point me in the right direction would be appreciated!

 

[jaseh86]

Hmm.. in the time I've had to think about this, I could only see one way to approach this, and it probably isn't the simplest approach.

As you said, you need to find the intersection between the projectile path and the line that represent the hill. A projectile takes the path of a parabola, while the hill would be a linear line. Do you know how to find the cartesian equation of a parabolic projectile path? (hint: substitute the t's so you only have y and x).

If you're stumped about the hill, just remember how to find the gradient of a line given its angle. Once you have TWO equations, it should just be a matter of algebra.