Derive an expression for the centripetal acceleration of a body in uniform circular motion.

(I assume familiarity with positions represented by vectors and differentiation of trigonometric functions). Consider the coordinates of a point moving in a circle of radius r around the origin. The equation of the circle is (rsin theta)² +(rcos theta)² = r². So the position vector x is (rcos theta; rsin theta) - this is a column vector. So differentiate with respect to time to get tangential velocity dx/dt: (-rsin thetadtheta/dt; rcos thetadtheta/dt). Differentiate again to get acceleration d²x/dt²: (-rcos theta(dtheta/dt)²-rsin thetad²theta/dt²; -rsin theta(dtheta/dt)²+rcos thetad²theta/dt²). Now dtheta/dt is of course constant since it's constant motion, which means d²theta/dt² = 0! So acceleration is now simply (-rcos theta(dtheta/dt)²; -rsin theta(dtheta/dt)²). We can relate velocity and position as |v|=r(dtheta/dt), and acceleration as a=|v|²/r.