Q:

Simple question about inertia law and trajectory

I know this question is so simple, maybe a middle school level, so if this question doesn't fit here, please let me know.
Suppose the particle bellow is in the vacuum with a constant velocity $\vec{v_1}$. If we add a force towards the left, we get an acceleration vector $\vec{a}$. Therefore, there is an increasing velocity $\vec{v_2}$ towards the left.
My question is $v_1$ is keeping pushing the particle up with the same speed forever? In this case how would be its trajectory?

A:

Yes, $\vec{v_{1}}$ (or $\vec{v_{y}}$) would be constant because there is no force accelerating nor decelerating along the direction of $\vec{v_{y}}$.
Therefore the equation of motion would be as follows, assuming upwards (along $\vec{v_{y}}$) is positive $y$ axis, and left (along $\vec{v_{x}}$) is negative $x$ axis.
$\vec{v_{y}} = v_{y} \vec{{j}}$ 
$\vec{v_{x}} = (-at) \vec{i}$
From the equations of motion, along $y$ axis, $y = {v_{y}}{t}$ -  (1)
And along $x$ axis, $x = -\frac{at^{2}}{2}$ - (2)
Eliminating $t$ from (1) and (2), we get $\frac{x}{y^{2}} = -\frac{a}{2(v_{y})^{2}}$, this is the equation of the path traveled by the particle.

