Why is centripetal acceleration towards the center?

To answer this, let's draw an object moving counter-clockwise in a circular path, and show its velocity vector at two different points in time.

Since we know is the rate of change of an object's velocity with respect to time, we can determine the direction of the object's acceleration by finding the direction of its change in velocity, Δv.

To find its change in velocity, Δv, we must recall that

Therefore, we need to find the difference of the vectors vf and vi graphically, which can be re-written as

Recall that to add vectors graphically, we line them up, tip-to-tail, and then draw our resultant vector from the starting point (tail) of our first vector to the ending point (tip) of our last vector.

So, the acceleration vector must point in the direction shown above. If I show this vector back on our original circle, lined up directly between our initial and final velocity vector, it's easy to see that the acceleration vector points toward the center of the circle.

You can repeat this procedure from any point on the circle... no matter where you go, the acceleration vector always points toward the center of the circle. In fact, the word centripetal in centripetal acceleration means "center-seeking!

(adapted from http://www.aplusphysics.com/courses/regents/circmotion/regents-circular-motion.html)