Q. There is a long hill with a gradient of 1:18 (3.18 degrees to the horizontal) on my way home from the shops. If I allow my car to coast down it, it maintains a constant speed of 55 m.p.h.

(A)My car's mass is 1350 kg. Assuming it has a rolling resistance R of 400 N, and an effective cross-sectional area of 2.2 m2, calculate its drag coefficient. (The density of air is 1.23 kg m-3. The rolling resistance, which arises through the deformation of the tyres, can be taken to be constant over the speed range of the car.)

(B)The maximum power output P of the engine is 120 kW. Show that if you ignore its rolling resistance, you can estimate the maximum speed of the car, on the flat and in still air, to be approximately 60 m s-1.

(C)Show that if you do take account of the rolling resistance of the car, the expression relating the maximum velocity v of the car to all the other quantities has the form:
A = Bv3 + Cv

(D) The expression found in part (d) is a cubic equation. There are standard algebraic solutions to this type of equation, but it is often quicker and easier to use other methods. Describe a way to get a reasonably accurate estimate of v (say, to the nearest 1 m s-1) without having to resort to practical experiments or manufacturer's data.

(E)The maximum air speed (as opposed to its ground speed) of the car in a headwind of 15 m s-1 is slightly higher than its maximum air speed in still air. Explain why this is the case.