Okay, so this is with respect to game design, so that’s where I’m coming from (please try to use smol words, I am no mathematician)
I have a 2D space ship. Its velocity is defined by vector A, let’s say (1i,0j), and the ship’s current position is (0,0). I also have a point P, let’s define it as (0,3). I’m trying to figure out how to select an angle at which the ship can be constantly accelerated at a rate of 1 unit/sec^2 in order to go from traveling along vector A to having a velocity that is directed exactly at point P. Ideally, this angle would result in the ship reaching the correct vector before it crosses point P in order for it to have time to turn around and negatively accelerate. I’m assuming I could find a way to plot a parabola using the calculus and stuff but it’s a little beyond me at the moment. Any help is appreciated :)
To answer this problem we will generalize the equation for acceleration in one direction. From kinematics, we know that: $$x_f = x_0 + v_{x0}\,t + \frac{1}{2}\,a_x\,t^2$$ Solving for acceleration, we find that: $$a_x = \frac{2\,(x_f - x_0 - v_{x0}\,t)}{t^2}$$ We will say that the initial position is always zero relative to the position of the ship, meaning that $x_0 = 0$. This simplifies our equation to: $$a_x = \frac{2\,(x_f - v_{x0}\,t)}{t^2} = 2\,x_f\,t^{-2} - 2\,v_{x0}\,t^{-1}$$ In order to find the optimal acceleration, we must find the critical points for $t$ by taking the time derivative: $$\frac{da_x}{dt} = -4\,x_f\,t^{-3} + 2\,v_{x0}\,t^{-2}$$ Looking at a graph, we can see that this is equal to zero only when $t = 2$ meaning: $$a_x = \frac{2\,[x_f - v_{x0}\,(2)]}{(2)^2} = \frac{x_f - 2\,v_{x0}}{2}$$ This equation is the optimal acceleration required in a single direction to get to a desired location where $x_f$ is the distance from the ship to the destination and $v_{x0}$ is the initial velocity in the same direction. To find the magnitude and direction of the acceleration vector in 2D space: $$a = \sqrt{a_x^2 + a_y^2}$$ $$\theta = \arctan{\left(\frac{a_y}{a_x}\right)}$$ You should always check that the angle makes sense with the components of your acceleration vector. If they do not match up, you can usually add $180^{\circ}$ to your angle value.