Question Concerning Magnitude And Direction Of The Accleration For Uniform Circular Motion To find the magnitude and direction of the acclleration for the uniform circular motion,we consider the below figure !enter image description here Where in particle $p$ moves at constant speed $v$ around a circle of radius $r$.At the instant shown,$p$ has coordinates $x_{p}$ and $y_{p}$. since velocity $\vec{v}$ of a moving particle is always tangent to the particle's path at the particle's position.In the above figure that means $\vec{v}$ is perpendicular to a radius $r$ drawn to the particle's position. > How does the angle $\theta$ that $\vec{v}$ makes with a verticle at $p$ equals the angle $\theta$ that radius $r$ makes with the $x$-axis?

The linear pair axiom does the job here.

Note that, the line marked $y_p$ is a straight line.

**Linear Pair Axioms:**

1. If a ray stands on line, then the sum of two adjacent angles so formed is $180^\circ$.
2. If the sum of two adjacent angles is $180^\circ$, then the non-common arms of the angles form a line.



Also, note that sum of the interior angles in a triangle is $180 ^\circ$.