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.