I know that the formula for centripetal acceleration is $\ a_c = v^2/r$.
Does the $v$ represent the average velocity of the car which is the arc length traveled divided by the time? If the answer is yes then, I have a second question. Why do we use distance divided by time to calculate the velocity instead of displacement. For example if there was a car that traveled a unbanked curve of radius 200m and completed a 2pi radian rotation in 10 sec. Find the centripetal acceleration. Why do we do $v= (2pi*200)/10$ instead of $v= (0)/10$ to find the $v$. Since $v$ should be displacement divided by time not distance.
[Physics] Is centripetal acceleration based on speed or velocity
accelerationkinematicsspeedvectorsvelocity
Best Answer
The conventions of differential calculus are subtle and hard to explain, and that seems to be the core of this conundrum. Note, 'velocity', a vector, is defined at every point of the trajectory of a moving object, and is defined as a limit of very-short-path-traveled divided by very-short-time-elapsed. So, the magnitude of the velocity, the speed, and the direction of the velocity (which, in circular motion, just takes on every value possible in two dimensions), are both important in general, but in the specific case of a complete circle, only magnitude has a non-trivial meaning. Expressing a formula with "v^2" reinforces that only the magnitude is intended, because there is no general way to 'square' a vector.
Acceleration, like velocity, is a vector defined at every point, by a similar limit of very-small-velocity-change divided by very-short-time-elapsed, and is inconsistent with using an AVERAGE of velocity (since it depends on the change over very short times). Acceleration has a direction just as velocity does, but (again, because this is a circle) it is described as 'centripetal' meaning it takes the direction toward the center of the circle.
It makes little sense to talk of average directed acceleration, because the full circular path makes that average vanish (by symmetry) in uniform circular motion: only the magnitude of the acceleration is constant in this kind of motion. So, the "centripetal acceleration" means only the magnitude of the instantaneous acceleration, without any averaging.
So, one cannot make use of the given formula while averaging vectors over a whole cycle of the circular motion. Differential calculus makes the hints, and therefore the formula for acceleration, more understandable.