[Physics] Maximum range of a projectile launched from elevation “dumbed down”

Question

I am trying to conceptually understand why the angle which produces the greatest range for a projectile launched with an elevation is not 45 degrees. I have exausted all other options, and I hope that someone here can explain it to me conceptually. That is, I am currently not at the mathematical level to explain this feature of projectiles using derivatives- but I am required to understand why this is the case.

The real source of my confusion is this app

http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html

The projectile in this application is launched from an elevation, and yet 45 degrees still produces the maximum range for the projectile! From my understanding the maximum range of projectile depends on configuring the launch to maximum amount of "hang-time." Is this application flawed – or am I going crazy.

Answer 1

As has already been mentioned in the other answers, a useful way to think about this problem is to consider the balance between the "hang time" you get from a vertical component of the velocity and the horizontal component that actually gets you some distance covered in however much "hang time" you have. What I'd like to add is a simple but powerful trick using very little mathematics* that I think may be helpful.

This trick is to consider a particular limiting case. If you imagine throwing a (soft, so nobody on the ground gets hurt) projectile from a very tall building, it is a pretty straightforward matter to convince yourself of the following: The additional "hang time" that you get from an upward component of the velocity will be small compared to the amount of time that the projectile will take to fall down the height of the tall building. This means that in order to cover the most ground, what you should really do is put all of the available energy into covering as much distance as possible, since you already have a large amount of "hang time" available to begin with. That is, the relative value of adding projectile time decreases.

Anyway, I hope this is a helpful way to think about it!

• I said very little mathematics, actually I did a bit of a trick, one that is useful in more circumstances than this. What I've done is called asymptotic analysis (examining the limiting behavior of the system). Often this can simplify the math quite a bit, and since the math wasn't too complicated to begin with, it's almost gone here!