Calculating an Interpolated Acceleration

Lake

I need to calculate my current acceleration from the following values:

initial_acceleration time_to max max_velocity current_velocity

Initial_acceleration is what we start with when accelerating from 0. Over the course of "time to max," acceleration follows linear interpolation from this value to whatever it needs to be to hit max_velocity at the end of this time. I can't use current_time to solve this problem because all these values are actually generated per frame based on other factors and any timer kept will become irrelevant next frame. I need to calculate what the timer should be from my min_velocity (0), current_velocity, and my max_velocity instead.

It sounds like I need something like an inverted quadratic interpolation, but I'm having trouble working it out.

Lake

func Solve_Acceleration(start_acceleration , max_speed, old_speed, duration, delta):
	var current_time = ...?
    	
	var new_time = (duration - current_time) + delta
	var min_part = lerp(0, start_acceleration * duration, new_time)
	var max_part = lerp(0, max_speed, new_time)
	var new_speed = lerp(min_part, max_part, new_time)
	return new_speed - old_speed

This is something like what I'm trying to accomplish, but I need to solve for current_time first. My understanding is that I need to do the inverse of what I've done here, but I don't know how. I'm not even sure whether this is a bilinear equation, or a quadratic equation.

I made an attempt at solving the problem as a quadratic equation but I got incorrect result:

var projection = start_acceleration * duration
var t = -projection - sqrt(pow(projection, 2) - (4 * speed * (max_speed - projection)))
t /= 2 * (max_speed - projection)