(* type used *)
type t_point = {x : float; y : float};;
type t_vector = {x : float; y : float};;
type t_ball = {position : t_vector; speed : t_vector};;

let sum_vector(vector_1, vector_2 : t_vector * t_vector) : t_vector =
   {x = vector_1.x +. vector_2.x; y = vector_1.y +. vector_2.y}
;;

let init_ball(height, speed : float * t_vector) : t_ball =
  let init_point : t_vector = {x = 0.; y = height} in {position = init_point; speed = speed}
;;

let gravity : t_vector = {x = 0.; y = -9.80665};;

let rec compute_next_speed(ball, t : t_ball * int) : t_vector =
  if t = 0
  then ball.speed
  else
    sum_vector(compute_next_speed(ball, t-1), gravity)
;;

let ball : t_ball = init_ball(10., {x = 5.; y = 50.});;

let rec compute_next_pos(ball , t: t_ball * int) : t_vector =
  if t = 0
  then ball.position
  else
    let speed1 : t_vector = compute_next_speed(ball, t-1) in
    let speed2 : t_vector = compute_next_speed(ball, t) in
    sum_vector(compute_next_pos(ball, t-1), {x = (speed1.x +. speed2.x) /. 2.; y = (speed1.y +. speed2.y) /. 2.})
;;

#trace compute_next_speed;;
#trace compute_next_pos;;

compute_next_pos(ball, 4);;

let rec compute_end_point(ball, t : t_ball * int) : t_vector =
  let pos : t_vector = compute_next_pos(ball, t) in
  if pos.y < 0.
  then compute_next_pos(ball, t)
  else
    compute_end_point(ball, t+1)
;;

compute_end_point(ball, 0);;
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