## Max Height and Time

I am a formal instructor and one of the classes I teach revolves
around the ballistics of a bullet. I am currently in a Calculus class
and will be starting as an Economics major In the fall when I finish
my time in the Marines. I am hoping to see if there is any way,
through calculus that I can prove the max height of the round through
actual mathematical proof. Many of the lesson plans and outlines
simply indentify what I have attached below and do not mention proof
of this. I am simply curious if I can prove the max height of the
round and at what time during its flight path does it reach this
height. Any help would be greatly appreciated. Below is a simple
diagram of a rifle trajectory at 300 yards. I attached the image to
give a preliminary view of what I am referencing.

Maximum Ordinate - The highest point in the trajectory of the round
on its route to the target
The event I would like to know if I can prove is how high the maximum
ordinate should be. I am new to calculus and especially the
application of calculus to real world events. Can I use a Derivative
function formula to find the max?

If so, what variables do I need for this, the rounds are all constant,
and with that the velocity of the round at specific ranges is also
fixed (in theory).
Would I use a function like h(t) = 4t^2+48t+3 to find the maximum h
and the corresponding time t. The conventional knowledge says that
roughly 2/3rds of the way to the target the projectile reaches its
maximum height. It is said to be roughly 7 inches above the line of
sight at its highest moment, which is supposedly 2/3rd of the way to 300 yards. This is what I am trying to prove or disprove throughcalculus.
All in all, any help or nudges to this type of problem would be
greatly appreciated.
 This seems like an appropriate problem for calculus, but I'm unsure of what assumptions are necessary to get the correct result. In other words, it's usually an introductory physics problem to treat the trajectory of an object under the influence of uniform gravity without drag; in that scenario, the maximum height always occurs at halfway through the trajectory. I suspect drag is needed for this problem, however. In other words, you'll have to solve a vector differential equation: $$\frac{d\vec p}{dt} = \vec F_{\text{drag}} + \vec F_{\text{gravity}}$$ The gravitational force is just going to be downward, but the drag will generally be in the opposite direction of the velocity. This is probably a tough problem to attack analytically, to be honest, which is why (I suspect) it's not often explicitly proven.
 Thanks for your response. I am going to have to do some research and reading to figure out a way to solve this. I have come across many issues with some of the material I teach that should be able to be proven mathematically. You mention that max height always occurs halfway through trajectory, but in this case all the literature teaches that 2/3rds of the way to the target a bullet will reach its max ordinate, or highest point.

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