# How to calculate the components of the displacement of unit magnitude of an accelerating particle in 2D? [closed]

A particle is moving in 2D having a constant acceleration $$\vec a$$.

Given initial velocity $$\vec u$$, after time $$t$$ the magnitude of its displacement $$\vec S$$ is 1.

I need to calculate $$S_x$$ and $$S_y$$ (components of the displacement in x and y direction) such that $$| \vec S | = 1$$, i.e. after time t.

I know $$\vec a$$ and $$\vec u$$, but I don't know 't'.

Is it possible to calculate the time 't' so that I can calculate $$S_x$$ and $$S_y$$ ?

I tried doing this to find 't' - $$\sqrt{ (u_x t + 0.5 a_xt^2)^2 + (u_yt + 0.5a_yt^2)^2} = 1 \\ \Longrightarrow\quad \left(\frac{a_x^2 + a_y^2}{4}\right)t^4 + (u_xa_x + u_ya_y)t^3 + (u_x^2 + u_y^2)t^2 = 1$$
However, without some numerical values for $$\vec a$$ and $$\vec u$$, this will be a monster to solve analytically (see here for quartic formula to see what I mean https://en.wikipedia.org/wiki/Quartic_function#/media/File:Quartic_Formula.svg).