Noobie question! I'm thrown off that the magnitude of velocity remains constant despite acceleration being non-zero. The change in direction of velocity makes sense to me, but wouldn't the magnitude of the vector change as well?
Let v = 5 m/s and r = 1 m.
mv^2 / r = ma -> a = v^2 / r. Does this imply that with acceleration = 5^2 / 1 = 25 m/s (toward the center of a circle), that velocity will change direction (for circular motion) but not magnitude?
While this is largely outside the scope of the MCAT, it can be worth elucidating to help you determine WHY UCM acts the way it does. Many physics students use Pythagoras theorem and a triangle to illustrate a force acting perpendicular to the velocity.
using this one would expect a different magnitude of velocity. However, when drawing this triangle, it assumes that the object has already moved in the direction of the force (maybe just a little bit). In doing so, we forget that the force then
stops being perpendicular.
What you have (re)discovered is simply that a force which is constant throughout space cannot remain perpendicular to the motion of a free particle if it lasts a finite amount of time: the particle will simply start to move in the direction of the force, picking up velocity as it starts moving in that direction.
However, if Force is not of the same magnitude and direction everywhere, one
can produce a force that remains perpendicular. The canonical example is that of a 2 bodies exerting gravity. The direction of the force is always towards the masses. If a test particle at a distance d moves past the other body with the proper velocity, this force will always remain perpendicular. Can you guess what kind of shape the trajectory of the test particle will trace out? you got it, a circle.
You're right that velocity has both magnitude AND direction...meaning a change in EITHER is considered a change in velocity and would require and outside force (and hence, resulting acceleration). Sure technically you might be able to detect small changes in velocity magnitude in real physics, but not for the UCM on the MCAT.
Uniform circular motion is a perfect example of this. At each instance, the object, say a satellite orbiting the earth, has a velocity tangential to its "circular" orbit and perpendicular to gravity (which would point to the center of the earth). While the magnitude of its velocity may stay the same, its direction is constantly being redirected by gravity to keep it moving in a circle and thus "accelerating" towards the earth. The same goes with spinning an object on a string, only the centripetal force is supplied by the tension in the string.
Hope this helps, good luck!
