In studying acceleration, the notion that an acceleration occurs when an object moves at constant speed in a circle is a difficult concept to master. If you have ever swung a ball on the end of string in a circle, it is clear that there is a force being provided by the hand to make the ball change direction as it moves. The centrally seeking force (the centripetal force), which is the tension in the string in this case, leads to a change in direction but not a change in speed. You see, you need to get behind or in front of the ball's motion if you want to change its speed. Tension cannot do that, since it is directly perpendicular to the ball's motion at every point on the circular path. A mathematical analysis of the vectors and and the limit as the change in motion becomes infinitesimal leads to the quantities that affect the acceleraion.
