Question

How does gravity relate to functions, specifically to s(t)= -1/2 gt^2 + vt + h.
Provide a brief history of the projectile and its use in functions.

Answer 1

projectile with no air friction and constant acceleration down of g

Vertical problem:
initial height = Hi
initial velocity component UPWARD = Vi
velocity upward = v
height = y
then
acceleration =-g which is about -9.81 m/s^2 on earth
v = Vi - g t
y = Hi + Vi t -(1/2)g t^2

That is all about up and down
There are no forces horizontal so the initial horizontal speed is forever until the crash.
call that speed u
then horizontal distance x = u t
===================
IF
you fire missile with speed S and angle up A
Then
Vi = S sin A
and
u = S cos A
the end :)

Answer 2

gravity provides a force of attraction between masses

a force acting on a mass causes it to accelerate

the function [s(t)= -1/2 gt^2 + vt + h] describes the height (s) relative to time (t) of a mass (object) in freefall
... g is the gravitational acceleration
... v is the initial vertical velocity
... h is the initial height

due to gravity, projectiles follow a parabolic
trajectory
... the horizontal velocity is constant
... while the vertical velocity changes due to gravity