Theory:

Please refer to page 67, Figure 3.11, in Serway's Physics for Scientists & Engineers, 3rd Edition. The car is moving on a circular path at constant speed. But its velocity is changing, since the direction is variable. Since the average acceleration is

we have to use vector subtraction of the velocities. Can you see how Serway proceeds to Fig. 3.11c from Fig. 3.11b in order to subtract the velocities? He's moved both velocity vectors to the same origin. To subtract, it's necessary to reverse the direction of v_i so its head is at the tail of v_f. Then the change in the velocity is drawn from the beginning of the two vectors (the tail of v_i to the head of the last vector v_f. Then he argues that the change in the velocity has the same direction as the acceleration (as you can see from the above equation) and that, in the limit as the time interval becomes very small, it points toward the center of the circle. Thus the acceleration points in the radial direction - toward the center of the circle.

On the next page, using similar triangles, Serway shows that the magnitude of the radial acceleration is

The tangential acceleration is computed from the change in the speed (or tangential speed) of the object. Serway's Figure 3.12 on page 69 shows this well. Notice that a_t is tangent to the path and a_r points toward the center? Also, when r is smaller, a_r is larger as in the middle part of the figure. The curvature changes from concave up to no curvature (where it says "Path of particle") to concave down. r is "infinity" where there is no curvature and the radial acceleration is zerothere.

Example: A car on a level curve