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

Question

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 $ ?

Please note :-

I tried doing this to find 't' - $$ \sqrt{ (u_x t + 0.5 a_xt^2)^2 + (u_yt + 0.5a_yt^2)^2} = 1 \\ \Longrightarrow\quad \left(\frac{a_x^2 + a_y^2}{4}\right)t^4 + (u_xa_x + u_ya_y)t^3 + (u_x^2 + u_y^2)t^2 = 1 $$

Is this the right approach? Is this equation really solvable for t ?

Answer 1

The equation you got is correct. The solution is the time at which the displacement magnitude is 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).

I would just paste that expression in the software of your choice and use the answer it gives.