a STEP-like mechanics question

I know the answer lol, I just thought it's a really nice question to share. The first part is meant to be a really easy warm up, it's the second part which is hard.

I've done this before, very nice. There's a method to do it with vectors as well, not my style but interesting none the less.

How did you do it without vectors?

How did you do it without vectors?

...

Sorry, I should've explained the question better, but the raised surface is flat, as is the ground which is below the raised surface, but the question you did looks interesting as well.

I added a picture to the original post.

I only properly read the question now. Yours is much more trivial, it's not really STEP like to be honest.

Oh ok, did you get the expression? I thought it'd be worthy of at least STEP 1. Try it and you might be surprised

4 am where I am, I'll do it in the morning, I'll probably be surprised at how hard I find it!

Did the question prove to be harder than you initially thought?

Forgot to do it, sorry. Just scribbled it onto a piece of paper, is it

$\displaystyle R = \frac{v \cos \theta}{g} \left(v\sin \theta + \sqrt{v^2 \sin^2 \theta + 2gh}\right)$

Then differentiate, set=0, etc...

Did the question prove to be harder than you initially thought? Added the answer in a spoiler.

To maximise:

$\displaystyle \theta = \arccos \sqrt{\frac{2gh + v^2}{2gh + 2v^2}}$

Hmm, that answer is equivalent to the one in the spoiler. How did you solve the resulting equation? I guess the question really was easy lol

I can't see your spoiler. I'm going out with my girlfriend, I'll be back in a few hours. Won't be able to reply till then, sorry!

Spoiler in the original post, your expression is equivalent though