## Max height of object

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 a formula 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? What variables do I need to find out at what range a bullet would reach its highest point over the line of sight?

If so, what variables do I need for this, the rounds are all constant, 62 gr bullets and the rifles are all the same.
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 through some form of math.
All in all, any help or nudges to this type of problem would be
greatly appreciated.

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 If you have a function like h(t) = 4t^2+48t+3 then you can take the derivitive of this function and find the values of t for which the derivitive is zero. Along with the two endpoints of your interval, these are the only places where a local maximum (or a local minimum) can occur. A proof of this is a simple consequence of the usual epsilon-delta definition of a derivitive. Your second step is then to evaluate h(t) at the endpoints and at each of the points that you have discovered where h'(t) = 0. The global maximum is at whichever t value gives the largest value for h(t). Your final step is to determine the position downrange that corresponds to the t value that you came up with. Since the round is not travelling at a constant velocity, this may not be a trivial exercise. If you were attacking this problem from first principles, you would not have a nice neat function describing the height of the round as a function of a parameter t. Instead you would probabaly have a formula for air resistance as a function of round velocity and orientation. Because you can relate force to acceleration (by F=ma), this would allow you to construct a "differential equation" relating the second derivitive of position with respect to time (acceleration) to the first derivitive of position with respect to time (velocity). Solving first order vector differential equations is probably not something you'll see in a first year calculus course.

 It is said to be roughly 7 inches above the line of sight at its highest moment,
You already have one equation and that is of linear translation.
You know the distance, and if you know the time of flight of the bullet, you can readly determine the drop of the bullet due to downwards acceleration due to gravity.