One of my favorite topics to teach is the idea of centripetal acceleration. I think this is one of my favorite topics because I was so confused about it as a physics student. Centripetal motion always meant “moving in a circle” to me, and I couldn’t understand how something like a pendulum at the bottom of its motion could be thought to be moving in a circle, when so clearly that isn’t the path we see.

In my class, we work to develop the idea that acceleration can be broken down into two perpendicular components, centripetal, which measures the acceleration associated with changing direction and tangential, which measures the acceleration associated with changing speed.

To get to this, I ask my students to think about all the ways we have of describing things that are perpendicular. What pairs of things can they come up with. Together, we think of two:

• x-and y-components of motion. By separating the x and y motion of a projectile, we find it’s much easier to comprehend its motion.
• Normal and frictional components of contact forces. If we think of contact forces as being made up of two independent forces—a normal force perpendicular to the surface, and frictional force parallel to the surface, its easier to reason about how contact forces affect the motion of an object.

Into this mix of strange perpendicular pairs, we add centripetal and tangential.

This then leads to the idea of how can we measure how much acceleration is associated with changing direction, and I ask my students to envision an object moving uniformly in circle, and the the pattern followed by the velocity vector, as shown in this crude illustration below.

From this, a student had a new insight to see that both the object itself, and the tip of the velocity vector trace out similar circles in the same time. From this, we can write:

$T=\frac{2\pi R}{v}$
and
$T=\frac{2\pi v}{a}$

Setting these two expressions equal gives
$\frac{2\pi R}{v}=\frac{2\pi v}{a}$

and solving for acceleration gives us
$a=\frac{v^2}{R}$

I was pretty blown away that a student came up with this completely on his own, and presented it to the class. Much better than my convoluted proof using arc lengths.

We discuss how, using this formula, we can now calculate the size of the centripetal acceleration of an object, but that it takes some thinking figure out the direction of this acceleration. Soon, students reason that it must always be in the direction that the velocity is changing, and so it’s perpendicular to the velocity vector. Centripetal acceleration measures the component of the acceleration perpendicular to the velocity vector, while tangential acceleration is the component of the acceleration parallel to the velocity vector. After some sketching of this on the whiteboards, things start to click together and students see that if the acceleration is always perpendicular to the velocity, that force does no work, and can’t change the kinetic energy of the object. Thus, in order for the object to speed up or slow down, there must be a component of the force parallel to the velocity. Everything fits together nicely.

After all this work, we study a bunch of the typical situations where objects experience centripetal motion, and students carefully label the forces acting on the object. It doesn’t take long to get into a battle about whether there should be a centripetal force acting on the object in a Free Body Diagra, and then I simply ask them to describe that force in our extended template (The [type of force] of the [object exerting force] on the [object experiencing force]). Suddenly, they see that while they can always find forces exerted by real objects that are responsible for the centripetal acceleration, there doesn’t seem to be some ever-present centripetal force that acts on things that change direction.

And so we set a rule—that the centripetal force should never be drawn on a FBD, just like the net force should never be drawn on the FBD (in fact, the centripetal force is just the direction changing component of the net force). Our generalized procedure for N2 analysis gains one question—”Is the object changing direction?” and if so, the requirement that you break acceleration down into centripetal and tangential components. Since students are seeing this far after they learned Newton’s Laws, it’s much easier to add one step to the procedure with out getting confused by the whole purpose of N2 analysis, as often happens when you throw in circular motion a week after students just saw blocks sliding down inclined planes.

With this framework, students are well prepared not just to deal with the simple uniform circular motion problems, but also to think about what happens with real situations, when you are slowing down around the banked curve, or the forces acting on a satellite in an elliptical orbit, like this vypthon simulation, where the yellow arrow is the velocity and he blue arrow shows the net force.

It’s a bit mind-boggeling to me that it took me half a dozen years in my own physics education to make this realization about centripetal motion.

Your student’s derivation of $a=\frac{v^2}{r}$ is how I’ve been teaching it ever since seeing it on the wikipedia page for it (http://en.wikipedia.org/wiki/Centripetal_force). I was trying to find a way to prove it without calculus so I was looking lots of places for various derivations. I really like that one the best and I’ve used it with the teachers in my licensure program. We take some time to talk about why we like it and it’s often part of the exam for that course.