2D Trajectory prediction in open space

Questions & Answers
1

I want to draw a line (I have Vectrocity) that shows predicted trajectory of an object based on acting forces (a single force to begin with).

Force is applied by this script:

using UnityEngine;
using System.Collections;

public class AccelerateToPoint : MonoBehaviour
{

    Rigidbody2D body;
    Transform myTransform;

    public float acceleration = 100;

    private float startTime;
    private bool move = false;
    private Vector3 targetPosition;

    // Use this for initialization
    void Start()
    {
        body = GetComponent<Rigidbody2D>();
        myTransform = transform;
    }

    // Update is called once per frame
    void FixedUpdate()
    {
        if (Input.GetMouseButtonDown(0)) {
            // Acquire destination point
            Vector3 pointer = Input.mousePosition;
            targetPosition = Camera.main.ScreenToWorldPoint(pointer);
            targetPosition.z = 0;
            move = true;
        }
        if (move) {
            Vector3 directionToTarget = (targetPosition - myTransform.position).normalized;
            // if target was changed mid-flight we have to find our velocity to the new target
            Vector3 velocityProjectionToTarget = Vector3.Project(body.velocity, directionToTarget);
            float q = velocityProjectionToTarget.normalized == directionToTarget ? 1 : -1;

            float S = (targetPosition - myTransform.position).magnitude;
            float t = Mathf.Sqrt(S * 2 / acceleration);
            // this is how much velocity we can reduce to zero on the way to our destination
            float V = acceleration * t;
            // this is our velocity to the target in a very near future
            float velocityThreshold = velocityProjectionToTarget.magnitude * q + (acceleration * Time.fixedDeltaTime * 2);



            bool accelerating;
            Vector3 forceDirection;
            if (V > velocityThreshold) {
                // if we can reduce more velocity than we have, we will accelerate
                accelerating = true;
                forceDirection = directionToTarget;
            } else {
                // otherwise we will decelerate
                forceDirection = -directionToTarget;
                accelerating = false;
            }

            // condition to stop the movement:
            // if we decelerate and we almost stopped - stop completely
            if (body.velocity.magnitude < 10 && !accelerating) {
                myTransform.position = targetPosition;
                body.velocity = Vector3.zero;
                move = false;
                return;
            }

            body.AddForce(forceDirection * acceleration);

            Debug.Log(
                 string.Format(
                     "t = {0} s; S = {1} m; V = {2} m/s; curV = {3}; {4}",
                     t, S, V, velocityProjectionToTarget.magnitude, accelerating
                 )
            );
        }
    }
}

If we just click and wait, object travels a straight line and stops at the destination.
However, if we change destination mid-flight, object starts orbiting around the destination point (orbit is usually elliptic and this ellipse usually rotates around destination point).

How to predict the motion?

2

I found my solution. You’ll need a bit of advanced maths.

These links provided all the necessary information to do what I need:

Algorithm described there is not demanding and it provides pretty accurate predictions.