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## Homework Statement

Alice and Bob are having a competition to see who can throw a baseball the farthest from the top of a hill which slopes downward uniformly at an angle

**ϕ**. Alice plays softball, and Bob doesn't have a particularly good arm, so he knows that he's probably gonna lose. Fortunately, Bob has taken mechanics, and he realizes that there is an optimal angle

**θ**at which he should throw the ball so that it has greatest range; what is that optimal angle?

Assume that Bob's initial velocity is fixed and the hill is infinitely tall and infinitely long, so that the ball always hits the hill.

**2. The attempt at a solution**

You don't have to look at my attempt to solve the equation, because it is fairly long.

My initial equations were:

x = v

_{0}cos(

**θ**)t

y = (v

_{0}cos(

**θ**)t)/tan(

**ϕ**) + v

_{0}sin(

**θ**)t - .5gt

^{2}

tan(

**ϕ**) = x/y

I solved for t in the x equation, and got:

t = x/(v

_{0}cos(

**θ**)

Then I substituted this value for t into the equation for y:

y = (v

_{0}cos(

**θ**)x)/tan(

**ϕ**)v

_{0}cos(

**θ**) + (v

_{0}sin(

**θ**)x)/(v

_{0}cos(

**θ**) - .5g(x/(v

_{0}cos(

**θ**))

^{2}

Then I substituted this value for y into the equation for tan(

**ϕ**).

tan(

**ϕ**) = x/(v

_{0}cos(

**θ**)x)/tan(

**ϕ**)v

_{0}cos(

**θ**) + (v

_{0}sin(

**θ**)x)/(v

_{0}cos(

**θ**) - .5g(x/(v

_{0}cos(

**θ**))

^{2}

Then I solved for x. After some rigorous simplification, x came out to be:

x = (2v

_{0}

^{2}(cos

^{2}(

**θ**) + sin(

**θ**)cos(

**θ**)tan(

**ϕ**))/(g)tan(

**ϕ**)

After that, I made a function of

**θ.**

f(

**θ**) = cos

^{2}(

**θ**) + (a)sin(

**θ**)cos(

**θ**), where a = tan(

**ϕ**)

x = (2v

_{0}

^{2}f(

**θ**))/tan(

**ϕ**)

x is maximized when f(

**θ**) is maximized. So I tried to find the derivative of f(

**θ**), set it equal to 0, and all that stuff, but I'm having trouble finding a value of

**θ**that maximizes f(

**θ**). Here is the derivative, just so you know:

f'(

**θ**) = -2sin(

**θ**)cos(

**θ**) +2(a)cos

^{2}(

**θ**) - a