To find the angle a particle makes with the horizontal at any time 't'

Question

Should you vector sum the position vectors at this time 't' or vector sum the velocity vector at this time 't' to find the angle a particle makes with the horizontal at any time 't'

Answer 1

It does not make sense to speak of the angle that a particle makes with a horizontal, since a point and a line together does not specify an angle.

Let $P(t)$ be the particle's position at time $t.$

Then, with respect to the origin $O$, the (acute) angle made at time $t$ by the particle with the horizontal is the acute angle between $\begin{pmatrix} 1 \cr 0 \cr 0 \end{pmatrix}$ and the position vector $\vec{OP}.$ (Since the acute angle is desired, when using dot product, remember to take its absolute value.)

On the other hand, the particle's direction of motion at time $t$ can be specified by the angle—not necessarily acute—between $\begin{pmatrix} 1 \cr 0 \cr 0 \end{pmatrix}$ and its velocity vector at time $t.$

If working in $\mathbb R^2$ instead of $\mathbb R^3,$ replace $\begin{pmatrix} 1 \cr 0 \cr 0 \end{pmatrix}$ with $\begin{pmatrix} 1 \cr 0 \end{pmatrix}.$