A car is traveling over the top of a hill, which we will model to be a piece of a circular arc of radius R. For each case below, draw careful to-scale motion maps and use them to construct the velocity vectors at the following three points: the top of the hill; equal time intervals on either side of the hill. Use the latter two velocity vectors to construct the average acceleration vector a at the top of the hill. Evaluate and comment on the direction and relative magnitude of a for the following cases:

Question

(a) The car is moving at constant speed v over a hill of radius R.

(b) The car is moving at constant speed 2v. Quantitatively, how does the magnitude a change compared to that in part (a)?

(c)The car is moving at constant speed v over a hill of radius R/2. Quantitatively, how does a now compare to that obtained in part (a)?

(d)Brownie points:From your results of parts (a)-(c), obtain an expression for the magnitude of the acceleration for the case of uniform circular motion (constant speed). Hint: what variables could the acceleration depend on? There is only one possible combination of those variables that will give you the right units!And your graphical constructions in parts (a)-(c) should confirm the functional dependences.

(e)The car speeds up as it goes over the top of the hill; the car slows down as it goes over the top of the hill. Discuss your results for this non-uniform circular motion in the context of the “superposition principle:” the idea that a seemingly complex motion (accelerating over a hill) can be broken down into the superposition (addition) of two simpler motions – perhaps better described here as a decomposition.