Describing motion

Let's look at the system you are describing from a very rigorous perspective: we have a dimensionless particle of mass $m$ moving along the real line with velocity $dot x(t)$. All of a sudden, a force $vec F(t)$ acts on the particle for an interval $delta t$, creating an angle $alpha$ with the real line.

For pure mathematical convenience, we describe the force as an impulse function like this

$$ F(t) = begin{cases} F_0cos{alpha} text{ if }tin delta t 0 text{ otherwise} end{cases}$$

According to Newton's Law: $F(t)=m {ddot x(t)}$ so we just need to integrate.

$$dot x(t) = frac{1}{m}int_{delta t} F(t)dt = frac{1}{m}F_0cos{alpha}delta t+v_0$$

Note that immediately after $delta t$ the force goes back to zero, meeaning the particle will proceded with constant velocity.